Almost Periodic and Almost Automorphic Functions in Abstract Spaces 2nd Edition Gaston M. N'Guérékata

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Gaston M. N'Guérékata
Almost Periodic
and Almost
Automorphic
Functions in
Abstract Spaces
SecondEdition

Almost Periodic and Almost Automorphic
Functions in Abstract Spaces

Gaston M. N’Guérékata
Almost Periodic and Almost
Automorphic Functions
in Abstract Spaces
Second Edition

Gaston M. N’Guérékata
Department of Mathematics
Morgan State University
Baltimore, MD, USA
ISBN 978-3-030-73717-7 ISBN 978-3-030-73718-4 (eBook)
https://doi.org/10.1007/978-3-030-73718-4
Mathematics Subject Classification: 34G10, 34G20, 34M03, 34K13, 34K30, 35B15, 35D30, 35F10,
35F20, 35F35, 35F50, 37C25, 39A12, 39A21, 39A23, 39A24, 45M15, 46A04, 45D05
1st edition: © Kluwer Academic Publishers 2001
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Switzerland AG 2021
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
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This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

This book is dedicated to my brother
the late
Maître François N’Guérékata

Preface to the First Edition
The aim of this monograph is to present for the first time a unified and homogeneous
exposition of the theory of almost automorphic functions and its application to the
fast growing field of differential equations in abstract spaces (Banach and Hilbert
spaces).
It is based essentially on the work of M. Zaki, S. Zaidman and the author during
the last three decades.
The concept of almost automorphy is a generalization of almost periodicity. It
has been introduced in the literature by S. Bochner in relation to some aspects
of differential equations [11–13], and [14]. Almost automorphic functions are
characterized by the following property:
Given any sequence of real numbers (s
n), we can extract a subsequence (sn) such
that
lim
n→∞
lim
m→∞
f (t + sn − sm) = f (t)
for each real number t. The convergence is simply pointwise while one requires
uniform convergence for almost periodicity.
In his important publication [67], W.A. Veech has studied almost automorphic
functions on groups. We like to mention the contribution by M. Zaki [70] which
provides a clear presentation of the study of almost automorphic functions with
values in a Banach space. Zaki’s work has been done under the supervision of
Professor S. Zaidman of the University of Montreal, Canada, and has since strongly
stimulated investigations in relation to the following problem:
What is the structure of bounded functions of the differential equation x =
Ax + f where f is an almost automorphic function?
This equation was originally raised and solved by Bohr and Neugebauer for an
almost periodic function f in a finite dimensional space. The generalization of this
result to the larger class of almost automorphic functions in infinite dimensional
spaces is not a trivial one. Indeed, it sometimes uses sophisticated techniques and
strong tools from functional analysis and operator theory.
vii

viii Preface to the First Edition
In this monograph we present several recent results from authors who contributed
to solve the above problem and consider some nonlinear cases. We deal with
classical solutions as well as the so-called mild solutions.
The concept of weak almost automorphy as presented by M. Zaki [70] is also
discussed (Chap. 2, Sect. 2).
Also, continuous solutions on the non-negative semi-axis that approach almost
automorphic functions at infinity are studied in Sections 5 and 6 of Chap. 2. In
particular semi-groups of linear operators are considered as an independent subject
in section of Chap. 3 and discussed in the context of the so-called Nemytskii and
Stepanov theory of dynamical systems.
A wide range of situations is presented in Chaps. 4 through 6.
In Chap. 3, we present some results of the theory of almost periodic functions
taking values in a locally convex space. We use a definition introduced in the
literature by C. Corduneanu and developed by the author for the first time in [54].
Applications to abstract differential equations are given in Chaps. 7 and 8. At the
end of each chapter, we have included a Notes section that gives some comments
the main references used.
It is our hope that this monograph will constitute a useful reference textbook for
post-graduate students and researchers in analysis, ordinary differential equations,
partial differential equations, and dynamical systems.
May it stimulate new developments of the theory of almost automotrphic and
almost periodic functions and enrich its applications to other fields.
It is a great pleasure to record our very sincere thanks to Professor Jerome
A. Goldstein, a friend and mentor for over two decades and Professor Georges
Anastassiou, who strongly encouraged us to complete this project.
We express my warm gratitude to Professor Constantin Corduneanu and Pro-
fessor Joseph Auslander for their valuable comments and suggestions. Our thanks
to our friend Professor Thomas Seidman who corrected some errors and Stephanie
Smith for her extraordinary skill and patience in setting this text.
We also express our appreciation to the editorial assistance of Kluwer Academic
Publishers, especially from Ana Bozicevic and Chris Curcio.
Finally, we owe a great deal to Professor Samuel Zaidman, who introduced us
to the exciting world of mathematical research. His experience and outstanding
contributions to mathematics have been a great source of inspiration to several
young mathematicians.

Preface to the Second Edition
Since the publication of our book [55] in 2001, there has been a real rebirth of the
theory of almost automorphic functions and applications to evolution equations as
we expected. An incredible number of researchers have been attracted by this topic.
This leads to a fast-growing number of publications.
We have received many helpful comments from colleagues and students, some
pointing out typographical errors, others asking for clarification and improvement
on some materials. In particular, Zheng, Ding, and N’Guérékata were able to
answer the long-time open problem: what is the “amount” of almost automorphic
functions which are not almost periodic in the sense of Bohr? The answer is
that the space of almost periodic functions is a set of first category in the space of
almost automorphic functions (cf. Chap. 1). Many other problems remain open, for
instance the study of almost periodic functions taking values in non-locally convex
spaces (cf. [30]).
Several generalizations were introduced in the literature including the study
of almost automorphic sequences. The interplay between almost automorphy and
almost periodicity is better known.
Researchers in the field overwhelmingly encouraged us to write a second edition
including some of the fresh and most relevant contributions and references.
As in the first edition, we present the materials in a simplified and rigorous way.
Each chapter is concluded with bibliographical notes showing the original sources
of the results and further reading.
We are most grateful to our numerous co-authors and colleagues who made such
great contributions to the theory of almost automorphy. We will not exhibit a list,
which would be any way incomplete, but we hope our friends will be satisfied with
our thanks and gratitude.
Finally, we thank our students Fatemeh Norouzi and Romario Gildas Foko
Tiomela and our friend and colleague Alexander Pankov for their careful proof-
reading and suggestions.
Baltimore, MD, USA Gaston M. N’Guérékata
October 2020
ix

Contents
1 Introduction and Preliminaries ........................................... 1
1 Banach Spaces ........................................................... 1
2 Lp Spaces ................................................................ 3
3 Linear Operators ......................................................... 5
4 Functions with Values in a Banach Space .............................. 6
5 Semigroups of Linear Operators ........................................ 8
6 Topological Vector Spaces .............................................. 9
7 The Exponential of a Bounded Linear Operator ....................... 10
8 Non-locally Convex Spaces ............................................. 15
2 Almost Automorphic Functions........................................... 17
1 Almost Automorphic Functions in a Banach Space.................... 17
2 Weak Almost Automorphy .............................................. 27
3 Almost Automorphic Sequences ........................................ 30
3.1 Kadets Theorem ................................................... 30
4 Asymptotically Almost Automorphic Functions ....................... 31
3 Almost Automorphy of the Function f (t, x) ............................ 37
1 The Nemytskii’s Operator ............................................... 37
4 Differentiation and Integration ........................................... 41
1 Differentiation in AA(X) ................................................ 41
2 Integration in AA(X) .................................................... 41
3 Differentiation in WAA(X) ............................................. 51
4 Integration in AAA(X) .................................................. 52
5 Pseudo Almost Automorphy............................................... 55
1 Pseudo Almost Automorphic Functions ................................ 55
2 μ-Pseudo Almost Automorphic Functions ............................. 58
6 Stepanov-like Almost Automorphic Functions .......................... 65
1 Definitions and Properties ............................................... 65
xi

xii Contents
7 Dynamical Systems and C0-Semigroups ................................. 71
1 Abstract Dynamical Systems............................................ 71
2 Complete Trajectories ................................................... 72
8 Almost Periodic Functions with Values in a Locally Convex Space... 79
1 Almost Periodic Functions .............................................. 79
2 Weakly Almost Periodic Functions ..................................... 84
3 Almost Periodicity of the Function f (t, x)............................. 97
4 Equi-Asymptotically Almost Periodic Functions ...................... 97
9 Almost Periodic Functions with Values in a Non-locally
Convex Space................................................................ 103
1 Definitions and Properties ............................................... 103
2 Weakly Almost Periodic Functions ..................................... 106
3 Applications.............................................................. 109
10 The Equation x’(t)=A(t)x(t)+f(t) .......................................... 111
1 The Equation x’(t)=A(t)x(t)+f(t) ........................................ 111
11 Almost Periodic Solutions of the Differential Equation in
Locally Convex Spaces ..................................................... 125
1 Linear Equations ......................................................... 125
1.1 The Homogeneous Equation x = Ax ............................ 126
1.2 The Inhomogeneous Case ......................................... 127
Appendix .......................................................................... 129
References......................................................................... 131

Chapter 1
Introduction and Preliminaries
This monograph presents several recent developments on the theory of almost
automorphic and almost periodic functions (in the sense of Bohr) with values in
an abstract space and its application to abstract differential equations. We suppose
that the reader is familiar with the fundamentals of Functional Analysis. However,
to facilitate the understanding of the exposition, we give in the beginning, without
proofs, some facts of the theory of topological vector spaces and operators which
will be used later in the text.
1 Banach Spaces
We denote by R and C the fields of real and complex numbers, respectively. We will
consider a (real or complex) normed space X, that is a vector space over the field
= R or C (respectively) with norm · .
Definition 1.1 A sequence of vectors (xn) in X is said to be a Cauchy sequence if
for every 0, there exists a natural number N such that xn − xm for all
n, m N.
Proposition 1.2 The following are equivalent:
(i) (xn) is a Cauchy sequence.
(ii) xnk+1 − xnk → 0 as k → ∞, for every increasing subsequence of positive
integers (nk).
Proposition 1.3 If (xn) is a Cauchy sequence in a normed space X, the sequence
of reals (xn) is convergent.
Definition 1.4 A Banach space X is a complete normed space, that is, a normed
space X in which every Cauchy sequence is convergent to an element of X.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
G. M. N’Guérékata, Almost Periodic and Almost Automorphic Functions in Abstract
Spaces, https://doi.org/10.1007/978-3-030-73718-4_1
1

2 1 Introduction and Preliminaries
Definition 1.5 A Banach space X is said to be uniformly convex if for every α, 0
α 2, there exists a number δ = δ(α) 0 such that for every x, y ∈ X with
x 1, y 1, x − y α, we have x + y ≤ 2(1 − δ).
Now if x, y ∈ X (not necessarily in the open unit ball), the conditions become

x + y
2

≤ (1 − δ) · max{x, y}
if
x − y ≥ α · max{x, y}.
We observe that Hilbert spaces are examples of uniformly convex Banach spaces.
Definition 1.6 A subset S of a normed space X is said to be open if for every x ∈ S,
there exists 0 such that the open ball
B(x, ) := {y ∈ X : x − y }
is included in S. S is said to be closed if its complement in X is open.
Proposition 1.7 A subset S of a normed space X is closed if and only if every
sequence of elements of S which converges in X, has its limit in S.
Definition 1.8 The closure of a subset S in a normed space X, denoted S, is the
intersection of all closed sets containing S.
It is easy to verify the following:
Proposition 1.9 Let S be a subset of a normed space X; then
S = {x ∈ X : ∃(xn) ⊂ S, lim
n→∞
xn = x}.
Definition 1.10 A subset S of a normed space X is said to be
(i) Dense in X if S = X;
(ii) Bounded in X if it is either empty or included in a closed ball;
(iii) Relatively compact in X if S is compact. Equivalently S is relatively compact if
and only if every sequence in S contains a convergent sequence. It is observed
that every relatively compact set is bounded.
Definition 1.11 Let X be a Banach space over the field = R or C. The
(continuous) dual space of X is the normed space of all bounded linear functionals
ϕ : X → which we denote X∗.
We can rewrite Definition 1.10 (ii) as follows:

2 Lp Spaces 3
Definition 1.12 A subset S of a Banach space X is said to be bounded if ϕ(S) is
bounded in for every ϕ ∈ X∗.
Proposition 1.13 ([54]) Weakly bounded sets are bounded in any Banach space X.
In particular every weakly convergent sequence is bounded in X.
We refer to (X∗)∗ = X∗∗, the bidual of X. X can be considered as embedded in
X∗∗ as follows:
For x ∈ X, let
J(x) : X∗
→ (= R or C)
be defined by
J(x)[ϕ] = ϕ(x), ϕ ∈ X∗
.
Then J(x) is a linear form. It is continuous since
|J(x)[ϕ]| = |ϕ(x)| ≤ ϕx|, ∀ϕ ∈ X∗
.
Hence J(x) ∈ X∗∗ for all x ∈ X. The map J : X → X∗∗ defined this way is also
linear and isometric. It is called the canonical embedding of X into its bidual X∗∗.
Definition 1.14 If the canonical embedding J : X → X∗∗ is surjective, i.e. X =
X∗∗, we say that X is reflexive.
Proposition 1.15 If X is a reflexive Banach space and (xn) is a bounded sequence,
then we can extract a subsequence (x
n) which will converge weakly to an element
of X.
2 Lp Spaces
Let I be an open interval of R and denote by Cc(I, X) the Banach space of all
continuous functions I → X with compact support.
Definition 1.16 A function f : I → X is said to be measurable if there exists a set
S ⊂ I of measure 0 and a sequence (fn) ⊂ Cc(I, X) such that fn(t) → f (t) for all
t ∈ I S.
It is clear that if f : I → X is measurable, then f : I → R is measurable too.
Theorem 1.17 Let fn : I → X, n = 1, 2, . . . be a sequence of measurable
functions and suppose that f : I → X and fn(t) → f (t), as n → ∞, for almost
all t ∈ I. Then f is measurable.
Proof We have fn → f on I S, where S is a set of measure 0. Let (fn.k) be a
sequence of functions in Cc(I, X) such that fn.k → f almost everywhere on I as

k → ∞. By Egorov’s Theorem (cf. [69, p. 16]) applied to the sequence of functions
fn,k−fn, there exists a set Sn ⊂ I of measure less that 1
2n such that fn,k−f → fn
uniformly on I Sn, as k → ∞.
Now let k(n) be such that fn,k(n) 1
n on I Sn and Fn := fn,k(n). Also let
B := S ∪ (∩m≥1

nm Sn). Then it is clear that B is a subset of I of measure 0.
Take t ∈ I B. So we get fn(t) → f (t), as n → ∞. On the other hand if n is large
enough, t ∈ I Sn. It follows that Fn − f 1
n , which means that Fn(t) → f (t),
as n → ∞, and consequently, f is measurable.

Remark 1.18 It is easy to observe that if φ : I → R and f : I → X are measurable,
then the product φf : I → X is measurable too.
Theorem 1.19 (Pettis Theorem) A function f : I → X is measurable if and only
if the following conditions hold:
(a) f is weakly measurable (i.e. for every x∗ ∈ X∗, the dual space of X, the
function x∗f : I → X is measurable).
(b) There exists a set S ⊂ I of measure 0 such that f (I S) is separable.
Proof See [69, p. 131].

We also have the following:
Theorem 1.20 If f : I → X is weakly continuous, then it is measurable.
Theorem 1.21 (Bochner’s Theorem) Assume that f : I → X is measurable.
Then f is integrable if and only if f is integrable. Moreover, we have

I
f

≤

I
f .
Proof Let f : I → X be integrable. Then there exists a sequence of functions
fn ∈ Cc(I, X), n = 1, 2, . . . such that

I fn(t) − f (t)dt → 0, as n → ∞. Using
the inequality f ≤ f − fn + fn, for all n, we see that f is integrable.
Conversely assume that f is integrable. Let Fn ∈ Cc(I, R), n = 1, 2, . . . be
a sequence of continuous functions such that

I |Fn − f | → 0 as n → ∞ and
|Fn| ≤ F almost everywhere for some F : I → R with

I |F| ∞.
Since f is measurable, there exists fn ∈ Cc(I, X), n = 1, 2, . . . such that
fn → f almost everywhere.
We now let
un :=
|Fn|
fn + 1
n
, n = 1, 2, . . .
Then it is obvious that un ≤ F, n = 1, 2, . . . and un → f almost everywhere on
I. Therefore

I un − f → 0 as n → ∞ and consequently f is integrable.
Using the Lebesgue–Fatou Lemma (cf. [69]), we get

3 Linear Operators 5

I
f

≤ lim
n→∞

I
un

≤

I
f .
This completes the proof.

Theorem 1.22 (Lebesgue’s Dominated Convergence Theorem) Let fn : I →
X, n = 1, 2, . . . be a sequence of integrable functions and g : I → R+ be an
integrable function. Let also f : I → X and assume that:
(i) for all n = 1, 2, . . . , fn ≤ g, almost everywhere on I.
(ii) fn(t) → f (t), as n → ∞ for all t ∈ I.
Then f is integrable on I and

I
f = lim
n→∞

I
fn.
Definition 1.23 Let 1 ≤ p ≤ ∞. We will denote by Lp(I, X) the space of all
classes of equivalence (with respect to the equality on I) of measurable functions
f : I → X such that f p is integrable. If we equip Lp(I, X) with the norm
f p :=

I
f (t)p
dt
1
p
, 1 ≤ p ∞
and
f ∞ := ess sup
I
f (t), p = ∞,
then Lp(I, X) turns out to be a Banach space.
We shall denote by L
p
loc(I, X) the space of all (equivalence classes of) measur-
able functions f : I → X such that the restriction of f to every bounded subinterval
of I is in Lp(I, X).
3 Linear Operators
Let us consider a normed space X and a linear operator A : X → X. We define the
norm of A by
|||A||| := sup
x=1
Ax.
Definition 1.24 A linear operator A : X → X is said to be continuous at x ∈ X
if for any sequence (xn) ⊂ X such that xn → x, we have Axn → Ax, that is,
Axn − Ax → 0 as xn − x → 0.

If A is continuous at each x ∈ Y ⊂ X, we say that A is continuous on Y.
Proposition 1.25 A linear operator A : X → X is continuous (on X) if and only if
it is continuous at a point of X.
Based on the above Proposition, we generally prove continuity of a linear operator
by checking its continuity at the zero vector.
Definition 1.26 A linear operator A : X → X is said to be bounded if there exists
M 0 such that Ax ≤ Mx for all x ∈ X.
We observe that a linear operator A : X → X is continuous if and only if it is
bounded.
Proposition 1.27 (The Uniform Boundedness Principle) Let F be a nonempty
family of bounded linear operators over a Banach space X. If sup
A∈F
Ax ∞ for
each x ∈ X, then sup
A∈F
|||A||| ∞.
Definition 1.28 A linear operator A in a normed space X is said to be compact if
AU is relatively compact, where U is the closed unit ball
U := {x ∈ X : x ≤ 1}.
Proposition 1.29 If X is a Banach space, the linear operator A : X → X
is compact if and only if for every bounded sequence (xn) ⊂ X, the sequence
(Axn) ⊂ X has a convergent subsequence; in other words, AS is relatively compact
for every bounded subset S of X.
4 Functions with Values in a Banach Space
We shall consider functions x : I → X where I is an interval of the real number set
R and X a Banach space.
Definition 1.30 A function x(t) is said to be (strongly) continuous at a point t0 ∈ I
if x(t) − x(t0) → 0 as t → t0 and strongly continuous on I if it is (strongly)
continuous at each point of I. If t0 is an end point of I, t → t0 (from the right or
from the left), accordingly.
x(t) is said to be weakly continuous on I if for any ϕ ∈ X∗, the dual space of X,
the numerical function (ϕx)(t) : I → R is continuous. It is obvious that the strong
continuity of x implies its weak continuity. The converse is not true in general.
In fact we have

4 Functions with Values in a Banach Space 7
Proposition 1.31 If x(t) : I → X is weakly continuous and has a range with a
compact closure in X, then x(t) is strongly continuous on I.
In this monograph, continuity will always denote strong continuity, unless otherwise
explicitly specified.
Proposition 1.32 Let I = [a, b]. Then the set C(I, X) of all continuous functions
x(t) : I → X is a Banach space when equipped with the norm
xC(I,X) := sup
t∈I
x(t).
Definition 1.33 A function x(t) : I → X is said to be differentiable at an interior
point t0 of I if there exists some y ∈ X such that x(t0+t)−x(t0)
t − y → 0 as
t → 0 and differentiable on an open subinterval of I if it is differentiable at each
point of I. Such y ∈ X, when it exists at t0 is denoted x(t0) and called the derivative
of x(t) at t0.
Definition 1.34 If the function x(t) : I → X is continuous on I = [a, b], we
define its integral on I (in the sense of Riemann) as the following limit:
lim
n→∞
n

k=1
x(tk)tk,
where the diameter of the partition a = t0 t1 . . . tn = b of I tends to zero.
When the limit exists we denote it by
b
a x(t)dt.
One can easily establish the estimate

b
a
x(t)dt

≤
b
a
x(t)dt.
Improper integrals are defined as in the case of classical calculus. For instance, if the
function is continuous on the interval [a, ∞), then we define its integral on [a, ∞)
as follows:
∞
a
x(t)dt = lim
b→∞
b
a
x(t)dt
if the limit exists in X. This integral is said to be absolutely convergent if
∞
a
x(t)dt ∞.

5 Semigroups of Linear Operators
Definition 1.35 Let A : X → X be a linear operator with domain D(A) ⊂ X, a
Banach space. The family T = (T (t))t≥0 of bounded linear operators on X is said
to be a C0-semigroup if
(i) For all x ∈ X, the mapping T (t)x : R+ → X is continuous.
(ii) T (t + s) = T (t)T (s) for all t, s ∈ R+ (semigroup property).
(iii) T (0) = I, the identity operator.
The operator A is called the infinitesimal generator (or generator in short) of the
C0-semigroup T if
Ax = lim
t→0+
T (t)x − x
t
and
D(A) := x ∈ X / lim
t→0+
T (t)x − x
t
exists .
It is observed that S commutes with T (t) on D(A). We define a C0-group in a
similar way, by replacing R+ by R.
For a bounded operator A, we have
T (t) := etA
=
∞

n=0
tnAn
n!
.
Theorem 1.36 Let T = (T (t))t≥0 be a C0-semigroup. Then there exists K ≥ 1
and α ∈ R such that
T (t) ≤ Keαt
, ∀t ≥ 0.
If α 0, we say that T is exponentially stable.
Proposition 1.37
(a) The function t → T (t) from R+ → R+ is measurable and bounded on any
compact interval of R+.
(b) The domain D(A) of its generator A is dense in X.
(c) The generator A is a closed operator.
For more details, cf. [35] and [69].

6 Topological Vector Spaces 9
6 Topological Vector Spaces
Let E be a vector space over the field ( = R or C). We say that E is a topological
vector space, which we denote E = E(τ), if E is equipped with a topology τ which
is compatible to the algebraic structure of E.
It is easy to check that for all a ∈ E, the translation f : E → E defined by
f (x) = x + a is a homeomorphism. Thus if is a base of neighborhoods of the
origin, + a is a base of neighborhoods of a. Consequently the whole topological
structure of E will be determined by a base of neighborhoods of the origin.
In this book, we will mainly use neighborhoods of the origin, which we
sometimes call neighborhoods in short.
Another interesting fact is that for every λ ∈ , λ = 0, the mapping f : E → E
defined by f (x) = λx is a homeomorphism, so that λU will be a neighborhood (of
the origin) if U is a neighborhood (of the origin), λ = 0.
Let us also recall the following:
Proposition 1.38 If is a base of neighborhoods, then for each U ∈ , we have:
(i) U is absorbing, that is for each x ∈ U, there exists λ 0 such that x ∈ αU
for all α with |α| ≥ λ;
(ii) There exists W ∈ such that W + W ⊂ U;
(iii) There exists a balanced neighborhood V such that V ⊂ U (A balanced or
symmetric set is a set V such that αV = V if |α| = 1).
A consequence of the above proposition is that every topological space E possesses
a base of balanced neighborhood.
We will call a locally convex topological vector space (or shortly a locally convex
space), every topological vector space which has a base of convex neighborhoods.
It follows that in a locally convex space, any open set contains a convex, balanced,
and absorbing open set.
A locally convex space whose topology is induced by an invariant complete
metric is called a Fréchet space.
Proposition 1.39 Let E be a vector space over the field ( = R or C). A function
p : E → R+ is called a seminorm if
(i) p(x) ≥ 0 for every x ∈ E;
(ii) p(λx) = |λ|p(x), for every x ∈ E and λ ∈ ;
(iii) p(x + y) ≤ p(x) + p(y), for every x, y ∈ E.
It is noted that if p is a seminorm on E, then the sets {x : p(x) λ} and
{x / p(x) ≤ λ}, where λ 0, are absorbing. They are also absolutely convex. We
recall that a set B ⊂ E is said to be absolutely convex if for every x, y ∈ E and
λ, μ ∈ , with |λ| + |μ| ≤ 1, we have λx + μy ∈ B.
Theorem 1.40 For every set Q of seminorms on a vector space E, there exists a
coarsest topology on E compatible with its algebraic structure and in which each

seminorm in Q is continuous. Under this topology, E is a locally convex space and
a base of neighborhoods is formed by the closed sets
{x ∈ E : sup
1≤i≤n
pi(x) ≤ },
where 0 and pi ∈ Q, i = 1, 2, . . . n.
Also E will be separated if and only if for each x ∈ E, x = 0, there exists a
seminorm p ∈ Q such that p(x) 0.
An important fact that will be used is the following consequence of the Hahn–
Banach Extension Theorem:
Proposition 1.41 ([69, page 107]) For each non-zero a in a locally convex space
E, there exists a linear functional ϕ ∈ E∗, the dual space of E, such that ϕ(a) = 0.
A subset S of a locally convex space is called totally bounded if, for every
neighborhood U, there are ai ∈ S, i = 1, 2, . . . n, such that
S ⊂ ∪n
i=1(ai + U).
It is clear that every totally bounded set is bounded. Also, the closure of a totally
bounded set is totally bounded.
We observe [69, page 13] that in a complete metric space, total boundedness and
relatively compactness are equivalent notions.
Now for functions of the real variable with values in a locally convex space E,
we define continuity, differentiability, and integration as in [54, 56, 69].
We finally revisit Proposition 1.27 in the context of locally convex spaces as
follows (cf. [45, page 199]):
Proposition 1.42 (Uniform Boundedness Principle) Let ϕ = {Aα : α ∈ }
where each Aα : E → F is a bounded linear operator and E, F are Fréchet
spaces. Suppose that {Aαx : α ∈ } is bounded for each x ∈ E. Then ϕ is
uniformly bounded.
Notes Details on this topic can be found in [66].
7 The Exponential of a Bounded Linear Operator
Let E be a complete, Hausdorff locally convex space.
Definition 1.43 A family of continuous linear operators Bα : E → E, α ∈ is
said to be equicontinuous if for any seminorm p, there exists a seminorm q such
that
p(Bαx) ≤ q(x), for any x ∈ E, any α ∈ .

7 The Exponential of a Bounded Linear Operator 11
Theorem 1.44 Let A : E → E be a continuous linear operator such that the family
{Ak : k = 1, 2, . . .} is equicontinuous. Then for each x ∈ E, t ≥ 0, the series
∞

k=0
tk
k!
Ak
x
(where A0 = I, the identity operator on E) is convergent.
Proof Let p be a seminorm on E. By equicontinuity of {Ak : k = 1, 2, . . .}, there
exists a seminorm q on E such that
p(Ak
x) ≤ q(x), for all k, and x ∈ E.
Therefore we have
p
m

k=n
tk
k!
Ak
x ≤
n

k=n
tk
k!
p(Ak
x) ≤ q(x)
m

k=n
tk
k!
,
which proves that the sequence
n

k=0
tk
k!
Ak
x is a Cauchy sequence in E. It is then
convergent and we denote the limit by
etA
x :=
∞

k=0
tk
k!
Ak
x.

Theorem 1.45 The mapping x → etAx, t ≥ 0, defines a continuous linear
operator E → E.
Proof Consider the linear operators An :=
n

k=0
tk
k!
Ak
, n = 0, 1, 2, . . . The family
{An : n = 0, 1, 2, . . .} is equicontinuous on any compact interval of R+.
Indeed, by equicontinuity of {Ak : k = 1, 2, . . .}, if p is a given seminorm, then
there exists a seminorm q such that
p(Anx) ≤
n

k=0
tk
k!
p(Ak
x) ≤ q(x)
n

k=0
tk
k!
≤ q(x)et
for every n = 0, 1, 2, . . .. It follows that
p(etA
x) ≤ q(x)et
,
for every t ≥ 0 and x ∈ E. This completes the proof.

Theorem 1.46 Let A and B be two continuous linear operators E → E such that
{An; n = 1, 2, . . .} and {Bn; n = 1, 2, . . .} are equicontinuous. Assume that A and
B commute, that is AB = BA; then
etA
· etB
= et(A+B)
, t ≥ 0.
Proof The proof is similar to the numerical case, that is for any real numbers a and
b, we have
∞

n=0
(ta)n
n!
.
∞

n=0
(tb)n
n!
=
∞

n=0
(t(a + b))n
n!
.
Indeed for any integer k and x ∈ E, we have
(A + B)k
x =
k

j=0

k
j

Aj
Bk−j
x =
k

j=0

k
j

Bk−j
Aj
x,
where k
j

= k!
j!(k−j)! .
In the last equality, we used the fact that AB = BA. Let p be a given seminorm
on E. Then there exists a seminorm q such that
p((A + B)k
x) ≤
k

j=0

k
j

p(Bk−j
Aj
x)
≤
k

j=0

k
j

q(Aj
x)
≤ 2k
sup
j≥0
q(Aj
x)
since
k

j=0

k
j

= 2k
.
This last inequality shows that the family

(A+B)k
2k : k = 1, 2, . . .

is equicon-
tinuous, so by Theorem (1.44), we can define et(A+B) by
et(A+B)
x :=
∞

n=0
(t(A + B))nx
n!
.
Now using the Cauchy product formula, we obtain

7 The Exponential of a Bounded Linear Operator 13
etA
· etB
=
∞

n=0
(tA)n
n!
·
∞

n=0
(tB)n
n!
=
∞

n=0
Cn,
where
Cn =
n

k=0
(tA)k
k!
·
(tB)n−k
(n − k)!
=
n

k=0
tn
k!(n − k)!
Ak
Bn−k
=
n

k=0

n
k

tn
n!
Ak
Bn−k
=
n

k=0
(t(A + B))n
n!
.
That means
∞

n=0
Cn = et(A+B)
. The proof is complete.

Theorem 1.47 Suppose that A is a continuous linear operator E → E such that
{An; n = 1, 2, . . .} is equicontinuous. Then for every x ∈ E, we have
lim
h→0+

ehA − I
h

x = Ax.
Proof Let p be a seminorm. Then there exists a seminorm q such that
p((ehA−I
h )x − Ax) = p(1
h (
∞

n=0
hn
n!
An
− I)x − Ax)
≤ p(1
h (
∞

n=2
hn
n!
An
x)
≤
∞

n=2
hn−1
n!
p(An
x)
≤ q(x)
∞

n=2
hn−1
n!
= q(x)

eh−1
h − 1

.
And since lim
h→+
eh − 1
h
= 1, we get the result.

From the above, we can deduce that

d
dt
etA
x = etA
· Ax = AetA
x,
Using the semigroup property above, we get also
e(t+s)A
= etA
· esA
.
We can use the same technique to prove similar results if t ≤ 0 and establish etA for
t ∈ R.
We are now ready to prove the following:
Theorem 1.48 The function etAx0 : R → E is the unique solution of the
differential equation
x
(t) = Ax(t), t ∈ R
satisfying x(0) = x0.
Proof Suppose there were another solution y(t) with y(0) = x0. Consider the
function v(s) = e(t−s)Ay(s), with t fixed in R; then we have
v(s) = −Ae(t−s)Ay(s) + e(t−s)Ay(s)
= −Ae(t−s)Ay(s) + e(t−s)AAy(s)
= 0,
for every s ∈ R. Therefore, v(s) = 0 on R, so that
v(t) = v(0), t ∈ R
or
y(t) = etA
y(0) = etA
x0, t ∈ R.
Since t is arbitrary, this completes the proof.

Let us recall the following fixed point theorem from [15]:
Theorem 1.49 Let D be a closed and convex subset of a Hausdorff locally convex
space such that 0 ∈ D, and let G be a continuous mapping of D into itself. If the
implication
(V = convG(V ), or V = G(V ) ∪ {0}) ⇒ V is relatively compact
holds for every subset V of D, then G has a fixed point.

8 Non-locally Convex Spaces 15
8 Non-locally Convex Spaces
It is well known that an F-space (X, +, ·, || · ||) is a linear space (over the field
= R or K = C) such that ||x +y|| ≤ ||x||+||y|| for all x, y ∈ X, ||x|| = 0 if and
only if x = 0, ||λx|| ≤ ||x||, for all scalars λ with |λ| ≤ 1, x ∈ X, and with respect
to the metric D(x, y) = ||x − y||, X is a complete metric space (see e.g. [25, p. 52],
or [37]). Obviously D is invariant to translations.
In addition, if there exists 0 p 1 with ||λx|| = |λ|p||x||, for all λ ∈ K, x ∈
X, then || · || will be called a p-norm and X will be called p-Fréchet space. (This
is only a slight abuse of terminology. Note that in e.g. [10] these spaces are called
p-Banach spaces). In this case, it is immediate that D(λx, λy) = |λ|pD(x, y), for
all x, y ∈ X and λ ∈ .
It is known that the F-spaces are not necessarily locally convex spaces. Three
classical examples of p-Fréchet spaces, non-locally convex, are the Hardy space
Hp with 0 p 1 that consists in the class of all analytic functions f : D → C,
D = {z ∈ C; |z| 1} with the property
||f || =
1
2π
sup
2π
0
|f (reit
)|p
dt; r ∈ [0, 1) +∞,
the sequences space
lp
=

x = (xn)n; ||x|| =
∞

n=1
|xn|p
∞

for 0 p 1, and the Lp[0, 1] space, 0 p 1, given by
Lp
= Lp
[0, 1] = f : [0, 1] → R; ||f || =
1
0
|f (t)|p
dt ∞ .
More generally, we may consider Lp( , , μ), 0 p 1, based on a general
measure space ( , , μ), with the p-norm given by ||f || =

|f |pdμ.
Some important characteristics of the F-spaces are given by the following
remarks:
Remark 1.50
(1) Three of the basic results in Functional Analysis hold in F-spaces too : the
Principle of Uniform Boundedness (see e.g. [25, p. 52]), the Open Mapping
Theorem, and the Closed Graph Theorem (see e.g. [37, p. 9–10]).
But on the other hand, the Hahn–Banach Theorem fails in non-locally convex
F-spaces. More exactly, if in an F-space the Hahn–Banach theorem holds, then
that space is necessarily locally convex space (see e.g. [37, Chapter 4]).

(2) If (X, +, ·, || · ||) is a p-Fréchet space over the field , 0 p 1, then its dual
X∗ is defined as the class of all linear functionals h : X → which satisfy
|h(x)| ≤ |||h||| · ||x||1/p, for all x ∈ X, where |||h||| = sup{|h(x)|; ||x|| ≤ 1}
(see e.g. [10, pp. 4–5]). Note that ||| · ||| in fact is a norm on X∗.
For 0 p 1, while (Lp)∗ = 0, we have that (lp)∗ is isometric to l∞—the Banach
space of all bounded sequences (see e.g. [37, p. 20–21]), therefore (lp)∗ becomes a
Banach space. Also, if φ ∈ (Hp)∗ then there exists a unique g, analytic on D and
continuous on the closure of D, such that
φ(f ) =
1
2π
lim
r→1
2π
0
f (reit
)g(e−it
)dt,
for all f ∈ Hp (see e.g. [26, p. 115, Theorem 7.5]). Moreover, (Hp)∗ becomes a
Banach space with respect to the usual norm |||φ||| = sup {|φ(f )|; ||f || ≤ 1} (see
the same paper [4]).
In both cases of lp and Hp, 0 p 1, their dual spaces separate the points of
corresponding spaces.
(3) The spaces lp and Hp, 0 p 1, have Schauder bases (see e.g. [37, p. 20],
for lp and [37, 64] for Hp). It is also worth to note that according to e.g. [28],
every linear isometry T of Hp onto itself has the form
T (f )(z) = α[φ
(z)]1/p
f (φ(z)), (8.1)
where α is some complex number of modulus one and φ is some conformal
mapping of the unit disc onto itself.

Chapter 2
Almost Automorphic Functions
1 Almost Automorphic Functions in a Banach Space
Definition 2.1 (S. Bochner [11–14]) Let X be a (real or complex) Banach space
and f ∈ C(R, X). We say that f is almost automorphic if for every sequence of
real numbers (s
n) there exists a subsequence (sn) such that
lim
m→∞
lim
n→∞
f (t + sm − sn) = f (t)
for each t ∈ R.
This is equivalent to the following:
Definition 2.2 f ∈ BC(R, X) is said to be almost automorphic if for every
sequence of real numbers (s
g(t) := lim
n→∞
f (t + sn)
exists for each t ∈ R and
lim
n→∞
g(t − sn) = f (t)
for each t ∈ R.
Remark 2.3
• The function g in Definition 2.2 is measurable, but not necessarily continuous.
• If the convergence in Definition 2.1 is uniform on compact subsets of R, then we
say that f is compact almost automorphic.
• If the convergence in Definitions 2.1 and 2.2 is uniform in t ∈ R, then f is almost
periodic. This shows that the class of almost automorphic functions is larger than
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
G. M. N’Guérékata, Almost Periodic and Almost Automorphic Functions in Abstract
Spaces, https://doi.org/10.1007/978-3-030-73718-4_2
17

18 2 Almost Automorphic Functions
the class of almost periodic functions. We will show later that the inclusion is
strict.
Theorem 2.4 ([27, 44, 58]) If the function g in the Definition 2.2 is continuous,
then f is uniformly continuous.
Proof Suppose f is not uniformly continuous on R. Then there exists a number
0 and two sequences (t
n) and (s
n) such that for each n, we have
f (s
n − t
n) − f (s
n)
and
lim
n→∞
t
n = 0.
In view of the almost automorphy of f , one can extract subsequences (sn) ⊂ (s
n)
and (sn + tn) ⊂ (s
n + t
n) such that
g1(t) := lim
n→∞
f (t + sn)
g2(t) := lim
n→∞
f (t + sn + tn)
for each t ∈ R.
Here g1 and g2 are continuous by assumption. Therefore we have
g1(0) − g2(0) .
Define the set
:= t ∈ R : g1(t) − g2(t)
3
4
.
Then is an open set in view of the continuity of g1 and g2. It is also nonempty
since 0 ∈ .
Now define for each n the set
An :=

t ∈ R : f (t + sm) − g1(t)≤

4
, f (t + sm + tm)−g2(t) ≤

4
, m≥n

.
Each An is nonempty, because of the convergence to g1 and g2 above. Now let
Bn := An ∩ , n = 1, 2, . . .
It is obvious that each Bn is nonempty (since 0 ∈ Bn), and

1 Almost Automorphic Functions in a Banach Space 19
∪∞
n=1Bn = .
Let n0 be large enough and take t ∈ Bn0 . Then t ∈ that means
() g1(t) − g2(t)
3
4
.
But since is open and g1 is continuous, we may choose m large enough, m ≥ n0
such that
t + tm ∈
and
g1(t + tm) − g1(t)

4
.
Since t ∈ Bn0 , we have
f (t + tm + sm) − g2(t) ≤

4
.
Also since = ∪∞
n=1Bn, there exists n1 such that t +tm ∈ Bn1 implies t +tm ∈ An1 .
So
f (t + tm + sm) − g1(t + tm) ≤

4
.
Finally, we obtain
g1(t) − g2(t) ≤ g1(t) − g1(t + tm) + g1(t + tm) − f (t + tm + sm)
+ f (t + tm + sm) − g2(t)
≤

4
+

4
+

4
=
3
4
,
which contradicts () and establishes the result.

Theorem 2.5 If f, f1, f2 are almost automorphic functions R → X and λ is a
scalar, then the following are true:
(i) λf and f1 + f2 are almost automorphic.
(ii) fa(·) := f (a + ·) is almost automorphic for every a ∈ R.
(iii) sup
t∈R
f (t) ∞.
(iv) The range Rf := {f (t) : t ∈ R} is relatively compact in X.
Proof Statements (i) and (ii) are obvious.

Let us prove (iii). Suppose by contradiction that sup
t∈R
f (t) = ∞. Then there
exists a sequence (s
n) of real numbers such that
lim
n→∞
f (s
n) = ∞.
Since f is almost automorphic, we can extract a subsequence (sn) ⊂ (s
n) such that
lim
n→∞
f (sn) = α
for some α ∈ R, that is
lim
n→∞
f (sn) = α ∞
which is a contradiction and establishes (iii).
(iv) Consider an arbitrary sequence (f (s
n)) in Rf . Since f is almost automor-
phic, we can extract a subsequence (sn) ⊂ (s
n) such that
lim
n→∞
f (sn) = g(0),
where g is the function in Definition 2.2. This proves that Rf is relatively compact
in X.

Remark 2.6 It is easy to observe that sup
t∈R
g(t) = sup
t∈R
f (t), which implies that
Rg = Rf .
Theorem 2.7 Let (fn) be a sequence of almost automorphic functions in a Banach
space X such that lim
n→∞
fn(t) = f (t) uniformly in t ∈ R.
Then f (t) is also almost automorphic.
Proof Let (s
n) be a sequence of real numbers. By the diagonal procedure, we can
extract a subsequence (sn) of (s
n) such that
lim
n→∞
fi(t + sn) = gi(t) (1.1)
for each i = 1, 2, . . . and each t ∈ R.
We claim that the sequence of functions (gi(t)) is a Cauchy sequence. Indeed if
we write
gi(t) − gj (t) = gi(t) − fi(t + sn) + fi(t + sn) − fj (t + sn)
+ fj (t + sn) − gj (t),
and use the triangle inequality, we get

gi(t) − gj (t) ≤ gi(t) − fi(t + sn) + fi(t + sn) − fj (t + sn)
+ fj (t + sn) − gj (t).
Let 0 be given. By uniform convergence of the sequence (fn), we can find a
natural number N such that for all i, j N,
fi(t + sn) − fj (t + sn) ,
for all t ∈ R and all n = 1, 2, . . . . Using Eq. (1.1) and the completeness of the
space X, we can deduce the pointwise convergence of the sequence (gi(t)), say to a
function g(t).
Let us prove that
lim
n→∞
f (t + sn) = g(t)
and
lim
n→∞
g(t − sn) = f (t)
pointwise on R.
Indeed, for each i = 1, 2, . . . , we get
f (t + sn) − g(t) ≤ f (t + sn) − fi(t + sn)
+ fi(t + sn) − gi(t) + gi(t) − g(t).
Given 0, we can find some natural number M such that
f (t + sn) − fM(t + sn) ≤
for every t ∈ R, n = 1, 2, . . . and gM(t) − g(t) for every t ∈ R, so that
f (t + sn) − g(t) ≤ 2 + fM(t + sn) − gM(t)
for every t ∈ R, n = 1, 2, . . . .
Now for every t ∈ R, we can find some natural number K depending on and
M such that
fM(t + sn) − gM(t)
for every n K.

Finally, we get
f (t + sn) − g(t) 3
for n ≥ N0 where N0 is some natural number depending on t and .
We can similarly prove that
lim
n→∞
g(t − sn) = f (t).

Let us denote by AA(X) (resp.AAc(X)) the space of all almost automorphic
functions (resp. compact almost automorphic) f : R → X. It turns out from the
above that AA(X) and AAc(X) are closed subspaces of BC(R, X). Thus they are
themselves Banach spaces under the supnorm
f AA(X) := sup
t∈R
f (t),
resp.
f AAc(X) := sup
t∈R
f (t).
If we denote by AP(X) the space of all almost periodic functions f : R → X (in
the sense of Bohr, cf. [22], or Chapter 4 below), then it is obvious that
AP(X) ⊂ AAc(X) ⊂ AA(X) (1.2)
and the inclusions are strict.
Let us state the following composition theorem:
Theorem 2.8 Let (X, · X), (Y, · Y) be Banach spaces over the same field ,
f ∈ AA(X) and Rf := {f (t) : t ∈ R} is the range of f . If φ : Rf → Y be a
continuous and bounded application, then the composite function φ ◦ f : Rf → Y
is also almost automorphic.
Proof Let (s
n) be sequence of real numbers. Since f ∈ AA(X), there exists a
subsequence (sn) such that
g(t) := lim
n→∞
f (t + sn)
lim
n→∞
g(t − sn) = f (t)
for each t ∈ R.

Since φ is continuous and bounded, we have
lim
n→∞
(φ ◦ f )(t + sn) = φ ◦ lim
n→∞
f (t + sn) = (φ ◦ g)(t)
lim
n→∞
(φ ◦ g)(t − sn) = φ ◦ lim
n→∞
g(t − sn) = (φ ◦ f )(t)
for each t ∈ R. This shows that φ ◦ f ∈ AA(X).

The following corollary follows immediately:
Corollary 2.9 If A is a bounded linear operator on X and f ∈ AA(X), then
(Af )(·) ∈ AA(X).
Let us now give some examples of almost automorphic functions which are not
almost periodic.
Example 2.10 (Levitan) Let f ∈ AP(R) and φ : Rf → Y be continuous and
bounded. Then φ ◦ f : Rf → Y may not be almost periodic. For example, let
x(t) = cost + cos
√
2t + 2
and
φ(s) = sin
1
s
.
Clearly φ(t) = sin

1
cost+cos
√
2t+2

is almost automorphic. But since φ(s) is not
uniformly continuous on Rx, then φ(t) is not almost periodic.
Example 2.11 (Veech) Consider the function x : R → C defined by
x(t) = eit
+ ei
√
2t
+ 2.
Let φ : C {0} → where is the unit circle in C be defined by
φ(x) =
x
|x|
.
Thus φ(t) =
eit + ei
√
2t + 2
|eit + ei
√
2t + 2|
is almost automorphic but not almost periodic.
Remark 2.12 If f ∈ AA(X) and φ : Rf ⊂ R → Y is not bounded, then φ◦f : Rf
in the proof of Theorem 2.8 is not well-defined for all t ∈ R, therefore φ ◦ f : Rf

is not almost automorphic. For example, replace φ(x) = sin1
x in the example above
by φ(x) = 1
x .
Theorem 2.13 ([72]) AP(X) is a set of first category in AA(X).
Proof It suffices to observe that AP(X) is a closed subset of AA(X) equipped with
the supnorm. Thus its interior is empty.

Theorem 2.14 Let T = (T (t))t∈R be a one parameter group of strongly continuous
linear operators uniformly bounded, i.e. there exists M 0 such that sup
t∈R
T (t) ≤
M. Let f ∈ AA(X) and S = f (Q), where Q denotes the set of rational numbers,
with the property that the function T (·)x ∈ AA(X) for each x ∈ S.
Then T (·)f (·) ∈ AA(X).
Proof Let B = {f (t) : t ∈ R} be the range of f . Then S is a countable and dense
subset of B.
Let S = (xn); then T (·)xn ∈ AA(X) for each n = 1, 2, . . . . Consider an arbitrary
n). Using the diagonal procedure, we can show that there
exists a subsequence (sn) of (s
n) such that
lim
n→∞
T (sn)x
exists for each x ∈ S. Pick x ∈ B. For any n, m, k we have
T (sn)x − T (sm)x ≤ T (sn)x − T (sn)xk
+ T (sn)xk − T (sm)xk
+ T (sm)xk − T (sm)x.
Therefore T (sn)xn − T (sm)xm → 0 since xn → S and we have
lim
n,m→∞
T (sn)x − T (sm)x ≤ 2Mx − xk.
Consequently, in view of the density of S in B, we can say that
lim
n→∞
T (sn)x
exists for every x ∈ B.
Now we observe that lim
n→∞
T (sn)x = y defines a mapping F from the linear
space spanned by B into X, namely
Fx = y if lim
n→∞
T (sn)x = y. (1.3)
The map F has the following properties:

(i) F is linear.
(ii) Fx = y ≤ lim
n→∞
T (sn)x ≤ Mx for every x in the subspace spanned
by B.
(iii) F is one-to-one.
(iv) If (xn) is a given sequence in B such that strong- lim
n→∞
xn = x exists, then
strong- lim
n→∞
T (sn)xn = Fx and strong- lim
n→∞
Fxn = Fx.
Let RF := {Fx : x ∈ B} be the range of F. Then we observe that
lim
n→∞
T (−sn)y
exists for every y ∈ RF .
It suffices to prove that
lim
n→∞
T (−sn)ym
exists for every ym ∈ F(S), where ym = F(xm), m = 1, 2, . . . .
Since T (t)xm ∈ AA(X) for each m = 1, 2, . . . , we have
lim
n→∞
T (t + sn)xm = lim
n→∞
T (t)T (sn)xm
= T (t) lim
n→∞
T (sn)xm
= T (t)Fxm
= T (t)ym
pointwise on R. Also we have
lim
n→∞
T (t − sn)ym = T (t)xm
= T (t) lim
n→∞
T (−sn)ym.
Now, for t = 0, we get
lim
n→∞
T (−sn)ym
exists for m = 1, 2, . . . and T (0)xm = xm. Hence, we get
lim
n→∞
T (−sn)y
exists for every y ∈ RF . This defines a linear map G on the linear subspace spanned
by RF where

Gy = lim
n→∞
T (−sn)y.
It is easy to verify that G has the same properties as F and we have
GFx = x
for every x ∈ B.
If (s
n) is an arbitrary sequence of real numbers, we can extract a subsequence
(sn) such that
lim
n→∞
f (t + sn) = g(t)
lim
n→∞
g(t − sn) = f (t)
pointwise on R and
lim
n→∞
T (−sn)x = y
exists for each x ∈ B.
Now let us observe that for every t ∈ R and n = 1, 2, . . . , we have
f (t + sn), g(t) ∈ B.
Let t ∈ R be arbitrary. Then for every n = 1, 2, . . .
T (t + sn)f (t + sn) = T (t)T (sn)f (t + sn)
so that
lim
n→∞
T (t + sn)f (t + sn) = T (t)Fg(t)
and
lim
n→∞
T (t − sn)Fg(t − sn) = T (t) lim
n→∞
T (−sn)Fg(t − sn)
= T (t)GFf (t).
The theorem is proved.

Theorem 2.15 Let f ∈ AA(X). If f (t) = 0 for all t α for some real number α,
then f (t) ≡ 0 for all t ∈ R.
Proof It suffices to prove that f (t) = 0 for t ≤ α. Consider the sequence of natural
numbers N = (n). By assumption there exists a subsequence (nk) ⊂ (n) such that

2 Weak Almost Automorphy 27
lim
k→∞
f (t + nk) = g(t)
lim
k→∞
g(t − nk) = f (t)
for each t ∈ R.
Obviously, for any t ≤ α, we can find (nkj ) ⊂ (nk) with t + nkj
α for all j = 1, 2, . . . , so that f (t + nkj ) = 0 for all j = 1, 2, . . . .
And since lim
j→∞
f (t+nkj ) = g(t), it yields g(t) = 0. Then we deduce that f (t) = 0.
The proof is complete.

Theorem 2.16 Let (T (t))t∈R be a C0-group and suppose that x(t) := T (t)x0 ∈
AA(X) for some x0 ∈ D(A), the domain of its infinitesimal generator A. Then
inf
t∈R
x(t) 0, or x(t) ≡ 0 f or every t ∈ R. (1.4)
Proof Assume that inf
t∈R
T (t)x0 = 0 and let (s
n) be a sequence of real numbers
such that lim
n→∞
x(s
n) = 0. We can extract a subsequence (sn) of (s
n) such that
lim
n→∞
x(t + sn) = y(t)
lim
n→∞
y(t − sn) = x(t)
for each t ∈ R. We have in fact
y(t) = lim
n→∞
T (t + sn)x0 = T (t) lim
n→∞
T (sn)x0 = T (t) lim
n→∞
x(sn) = 0
for each t ∈ R. We deduce that x(t) ≡ 0 on R, and the proof is complete.

2 Weak Almost Automorphy
Definition 2.17 A weakly continuous function f : R → X is said to be weakly
almost automorphic (in short w-almost automorphic) if for every sequence of real
numbers (s
weak − lim
n→∞
f (t + sn) = g(t)

weak − lim
n→∞
g(t − sn) = f (t)
for each t ∈ R.
Remark 2.18
(i) Every almost automorphic function is w-almost automorphic.
(ii) If f : R → X is w-almost automorphic, then the function F : R → R defined
by F(t) := (ϕf )(t) with ϕ ∈ X∗ the dual space of X is almost automorphic.
The following results are obvious and we omit the proof:
Theorem 2.19 If f, f1, f2 are w-almost automorphic, then the following also are
w-almost automorphic:
(i) f1 + f2.
(ii) cf for an arbitrary scalar c.
(iii) fa(t) := f (t + a), for any fixed real number a.
We denote by WAA(X) the vector space of all w-almost automorphic functions
f : R → X.
Theorem 2.20 If f ∈ WAA(X), then sup
t∈R
f (t) ∞.
Proof Suppose by contradiction that sup
t∈R
f (t) = ∞. Then there exists a
n) such that lim
n→∞
f (s
n) = ∞. Since f is w-almost
automorphic, then we can find a subsequence (sn) such that
weak − lim
n→∞
f (sn) = α exists.
(f (sn)) is then a weakly convergent sequence, hence it is weakly bounded and
therefore bounded by Proposition 1.41. This is a contradiction, and consequently,
the theorem holds.

Theorem 2.21 If f ∈ WAA(X), then
sup
t∈R
f (t) = sup
t∈R
g(t),
where g is the function defined in Definition 2.17.
Proof Since every weakly convergent sequence is bounded in norm (Proposi-
tion 1.41), and in particular if
weak − lim
n→∞
xn = α,

2 Weak Almost Automorphy 29
then
α ≤ lim inf
n→∞
xn
(cf. [69, Theorem 1, page 120]). Thus, for each t ∈ R, we get
g(t) ≤ lim inf
n→∞
f (t + sn) ≤ sup
t∈R
f (t) ∞
and
f (t) ≤ lim inf
n→∞
g(t − sn) ≤ sup
t∈R
g(t) ∞.
The equality is now proved.

The following result is easy to prove:
Theorem 2.22 Let f ∈ WAA(X) and A ∈ B(X). Then Af : R → X is also
w-almost automorphic.
Theorem 2.23 Let f ∈ WAA(X) and suppose that its range Rf is relatively
compact in X. Then f ∈ AA(X).
Proof Let (s
n) be a sequence of real numbers. We can extract a subsequence (sn) ⊂
(s
n) such that
weak − lim
n→∞
f (t + sn) = g(t)
weak − lim
n→∞
g(t − sn) = f (t)
for each t ∈ R. Now fix t0 ∈ R. Then we have
lim
n→∞
(ϕf )(t0 + sn) = (ϕg)(t0)
and
lim
n→∞
(ϕg)(t0 − sn) = (ϕf )(t0)
for every ϕ ∈ X∗.
Observe that the range Rg of g is also relatively compact in X.
Indeed, for every ¯
t ∈ R, g(¯
t) is the strong limit of the sequence (f (¯
t +sn)) which
is contained in the closure of Rf ; whence g(¯
t) is in the closure of Rf , a compact
set in X.

Also from the weak convergence of the sequence (g(¯
t − sn)) toward f (¯
t), for
every ¯
t ∈ R, we have the strong convergence, so f ∈ AA(X).

3 Almost Automorphic Sequences
Similarly as for functions, we define below the almost automorphy of sequences.
From now on, we will use the notation l∞(X) to indicate the space of all bounded
(two-sided) sequences in a Banach space X with supnorm, that is, if x = (xn)n∈Z ∈
l∞(X), then
x := sup
n∈Z
xn.
Definition 2.24 A sequence x ∈ l∞(X) is said to be almost automorphic if for any
sequence of integers (k
n), there exists a subsequence (kn) such that
lim
m→∞
lim
n→∞
xp+kn−km = xp (3.1)
for any p ∈ Z.
The set of all almost automorphic sequences in X forms a closed subspace
of l∞(X), that is denoted by aa(X). We can show that the range of an almost
automorphic sequence is precompact. For each bounded sequence g := (gn)n∈Z
in X, we will denote by S(k)g the k-translation of g in l∞(X), i.e., (S(k)g)n =
gn+k, ∀n ∈ Z. And S stands for S(1).
3.1 Kadets Theorem
Let c0 be the Banach space of all numerical sequences (an)∞
n=1 such that lim
n→∞
an =
0, equipped with supnorm. In the simplest case, the problem we are considering
becomes the following:
when is the integral of an almost automorphic function also almost auto-
morphic?
We can take the same counterexample as in [38] to show that additional
conditions should be imposed on the space X.
Example 2.25 Consider the function f (t) with values in c0 defined by
f (t) = ((1/n) cos(t/n))∞
n=1, ∀t ∈ R.

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Almost Periodic and Almost Automorphic Functions in Abstract Spaces 2nd Edition Gaston M. N'Guérékata

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Robert Blincoe, an Orphan Boy

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Transcriber’s Note: Obvious printing errors have been corrected, but old spelling (e.g.
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A
MEMOIR
OF
ROBERT BLINCOE,
An Orphan Boy;
SENT FROM THE WORKHOUSE OF ST.
PANCRAS, LONDON,
AT SEVEN YEARS OF AGE,
TO ENDURE THE
Horrors of a Cotton-Mill,
THROUGH HIS INFANCY AND YOUTH,
WITH A MINUTE DETAIL OF HIS
SUFFERINGS,
BEING
THE FIRST MEMOIR OF THE KIND
PUBLISHED.
BY JOHN BROWN.
MANCHESTER:
PRINTED FOR AND PUBLISHED BY J.
DOHERTY, 37, WITHY-GROVE,
1832.

PUBLISHER’S PREFACE.
The various Acts Of Parliament, which have been passed, to
regulate the treatment of children in the Cotton Spinning
Manufactories, betoken the previous existence of some treatment,
so glaringly wrong, as to force itself upon the attention of the
legislature. This Cotton-slave-trade, like the Negro-slave-trade, did
not lack its defenders, and it might have afforded a sort of sorry
consolation to the Negro slaves of America, had they been informed,
that their condition, in having agriculturally to raise the cotton, was
not half so bad, as that of the white infant-slaves, who had to assist
in the spinning of it, when brought to this country. The religion and
the black humanity of Mr. Wilberforce seem to have been entirely of
a foreign nature. Pardon is begged, if an error is about to be
wrongfully imputed—but the Publisher has no knowledge, that Mr.
Wilberforce’s humane advocacy for slaves, was ever of that homely
kind, as to embrace the region of the home-cotton-slave-trade. And
yet, who shall read the Memoir of Robert Blincoe, and say, that the
charity towards slaves should not have begun or ended at home?
The Author of this Memoir is now dead; he fell, about two or three
years ago, by his own hand. He united, with a strong feeling for the
injuries and sufferings of others, a high sense of injury when it bore
on himself, whether real or imaginary; and a despondency when his
prospects were not good.—Hence his suicide.—Had he not
possessed a fine fellow-feeling with the child of misfortune, he had
never taken such pains to compile the Memoir of Robert Blincoe, and
to collect all the wrongs on paper, on which he could gain
information, about the various sufferers under the cotton-mill
systems. Notes to the Memoir of Robert Blincoe were intended by

the author, in illustration of his strong personal assertions. The
references were marked in the Memoir; but the Notes were not
prepared, or if prepared, have not come to the Publisher’s hand. But,
on inquiring after Robert Blincoe, in Manchester, and mentioning the
Memoir of him written by Mr. Brown, as being in the Publisher’s
possession, other papers, by the same Author, which had been left
on a loan of money in Manchester, were obtained, and these papers
seem to have formed the authorities, from which the Notes to the
Memoirs would have been made. So that, though the Publisher does
not presume to make notes for the Author, nor for himself, to this
Memoir, he is prepared to confirm much of the statement here
made, the personalities of Robert Blincoe excepted, should it be
generally challenged.
Robert Blincoe, the subject of the Memoir, is now about 35 years
of age, and resides at No. 19, Turner-street, Manchester, where he
keeps a small grocer’s shop. He is also engaged in manufacturing
Sheet Wadding and Cotton Waste-Dealer. The Publisher having no
knowledge of Robert Blincoe, but in common with every reader of
this Memoir, can have no personal feelings towards him, other than
those of pity for his past sufferings. But such a Memoir as this was
much wanted, to hand down to posterity, what was the real
character of the complaints about the treatment of children in our
cotton mills, about which a legislation has taken place, and so much
has been said. An amended treatment of children has been made,
the apprenticing system having been abandoned by the masters of
the mills; but the employment is in itself bad for children—first, as
their health—and second, as to their manners and acquirements—
the employment being in a bad atmosphere; and the education,
from example, being bad; the time that should be devoted to a
better education, being devoted to that which is bad. The
employment of infant children in the cotton-mills furnishes a bad
means to dissolute parents, to live in idleness and all sorts of vice,

upon the produce of infant labour. There is much of this in
Lancashire, which a little care and looking after, on the part of the
masters of cotton-mills, might easily prevent. But what is to be
done? Most of the extensive manufacturers profit by human misery
and become callous toward it; both from habit and interest. If a
remedy be desired, it must be sought by that part of the working
people themselves, who are alive to their progressing degradation. It
will never be sought fairly out, by those who have no interest in
seeking it. And so long as the majority of the working people
squanders its already scanty income in those pest-houses, those
intoxicating nurseries, for vice, idleness and misery, the public
drinking-houses, there is no hope for them of an amended condition.
MEMOIR
OF
Robert Blincoe,
AN ORPHAN BOY.

CHAP. I.
By the time the observant reader has got through the melancholy
recital of the sufferings of Blincoe and his associates in cotton-mill
bondage, he will probably incline to an opinion, that rather than rear
destitute and deserted children, to be thus distorted by excessive
toil, and famished and tortured as those have been, it were
incomparably less cruel to put them at once to death—less cruel that
they had never been born alive; and far more wise that they had
never been conceived. In cases of unauthorized pregnancies, our
laws are tender of unconscious life, perhaps to a faulty extreme;
whilst our parochial institutions, as these pages will prove, after
incurring considerable expence to preserve the lives of those forlorn
beings, sweep them off by shoals, under the sanction of other legal
enactments, and consign them to a fate, far worse than sudden
death.
Reared in the most profound ignorance and depravity, these
unhappy beings are, from the hour of their birth, to the last of their
existence, generally cut off from all that is decent in social life. Their
preceptors are the veriest wretches in nature!—their influential
examples all of the worst possible kind. The reports of the Cotton Bill
Committees abundantly prove, that, by forcing those destitute poor
to go into cotton-mills, they have, in very numerous instances, been
consigned to a destiny worse than death without torture. Yet
appalling as are many of the statements, which, through the reports
of the Committees, have found their way before the public, similar
acts of delinquencies, of a hue still darker—even repeated acts of
murder, have escaped unnoticed. Much of the evidence brought
forward by the friends of humanity, was neutralized or frittered away
by timidity of their witnesses, or by the base subserviency of venally
unprincipled professional men, who, influenced by rich capitalists,
basely prostituted their talent and character as physicians, surgeons,

and apothecaries, to deceive the government, to perplex and
mislead public opinion, and avert the loud cry raised against the
insatiate avarice and relentless cruelty of their greedy and unfeeling
suborners.
It was in the spring of 1822, after having devoted a considerable
time to the investigating of the effect of the manufacturing system,
and factory establishments, on the health and morals of the
manufacturing populace, that I first heard of the extraordinary
sufferings of R. Blincoe. At the same time, I was told of his earnest
wish that those sufferings should, for the protection of the rising
generation of parish children, be laid before the world. Thus
assured, I went to enquire for him, and was much pleased with his
conversation. If this young man had not been consigned to a cotton-
factory, he would probably have been strong, healthy, and well
grown; instead of which, he is diminutive as to stature, and his
knees are grievously distorted. In his manners, he appeared
remarkably gentle; in his language, temperate; in his statements,
cautious and consistent. If, in any part of the ensuing narrative,
there are falsehoods and misrepresentations, the fault rests solely
with himself; for, repeatedly and earnestly, I admonished him to
beware, lest a too keen remembrance of the injustice he had
suffered should lead him to transgress the limits of truth. After I had
taken down his communications, I tested them, by reading the same
to other persons, with whom Blincoe had not had any intercourse on
the subject, and who had partaken of the miseries of the same hard
servitude, and by whom they were in every point confirmed.
Robert Blincoe commenced his melancholy narrative, by stating,
that he was a parish orphan, and knew not either his father or
mother. From the age of four years, he says, “till I had completed
my seventh, I was supported in Saint Pancras poorhouse, near
London.” In very pathetic terms, he frequently censured and
regretted the remissness of the parish officers, who, when they
received him into the workhouse, had, as he seemed to believe,
neglected to make any entry, or, at least, any to which he could
obtain access, of his mother’s and father’s name, occupation, age, or

residence. Blincoe argued, and plausibly too, that those officers
would not have received him, if his mother had not proved her
settlement; and he considered it inhuman in the extreme, either to
neglect to record the names of his parents, or, if recorded, to refuse
to give him that information, which, after his attaining his freedom,
he had requested at their hands. His lamentations, on this head,
were truely touching, and evinced a far higher degree of
susceptibility of heart, than could have been expected from the
extreme and long continued wretchedness he had endured in the
den of vice and misery, where he was so long immured. Experience
often evinces, that, whilst moderate adversity mollifies and expands
the human heart, extreme and long continued wretchedness has a
direct and powerful contrary tendency, and renders it impenetrably
callous.
In one of our early interviews, tears trickling down his pallid
cheeks, and his voice tremulous and faltering, Blincoe said, “I am
worse off than a child reared in the Foundling Hospital. Those
orphans have a name given them by the heads of that institution, at
the time of baptism, to which they are legally entitled. But I have no
name I can call my own.” He said he perfectly recollected riding in a
coach to the workhouse, accompanied by some female, that he did
not however think this female was his mother, for he had not the
least consciousness of having felt either sorrow or uneasiness at
being separated from her, as he very naturally supposed he should
have felt, if that person had been his mother. Blincoe also appeared
to think he had not been nursed by his mother, but had passed
through many hands before he arrived at the workhouse; because
he had no recollection of ever having experienced a mother’s
caresses. It seems, young as he was, he often enquired of the
nurses, when the parents and relations of other children came to see
his young associates, why no one came to him, and used to weep,
when he was told, that no one had ever owned him, after his being
placed in that house. Some of the nurses stated, that a female, who
called soon after his arrival, inquired for him by the name of “Saint;”
and, when he was produced, gave him a penny-piece, and told him

his mother was dead. If this report were well founded, his mother’s
illness was the cause of his being removed and sent to the
workhouse. According to his own description, he felt with extreme
sensibility the loneliness of his condition, and, at each stage of his
future sufferings, during his severe cotton-mill servitude, it pressed
on his heart the heaviest of all his sorrows—an impassable barrier, “a
wall of brass,” cut him off from all mankind. The sad consciousness,
that he stood alone “a waif on the world’s wide common;” that he
had no acknowledged claim of kindred with any human being, rich
or poor—that he stood apparently for ever excluded from every
social circle, so constantly occupied his thoughts, that, together with
his sufferings, they imprinted a pensive character on his features,
which probably neither change of fortune, nor time itself, would ever
entirely obliterate. When he was six years old, and, as the
workhouse children were saying their Catechism, it was his turn to
repeat the Fifth Commandment—“Honour thy father and thy mother,
c.,” he recollects having suddenly burst into tears, and felt greatly
agitated and distressed—his voice faltering, and his limbs trembling.
According to his statement, and his pathetic eloquence, in reciting
his misfortunes, strongly corroborated his assertion, he was a very
ready scholar, and the source of this sudden burst of grief being
inquired into by some of his superiors, he said, “I cry, because I
cannot obey one of God’s commandments, I know not either my
father or my mother, I cannot therefore be a good child and honour
my parents.”
It was rumoured, in the ward where Robert Blincoe was placed,
that he owed his existence to the mutual frailties of his mother and a
reverend divine, and was called the young Saint, in allusion to his
priestly descent. This name or appellation he did not long retain, for
he was afterwards called Parson; often, the young Parson; and he
recollected hearing it said in his presence, that he was the son of a
parson Blincoe. Whether these allusions were founded in truth, or
were but the vile effusions of vulgar malice, was not, and is not, in
his power to determine, whose bosom they have so painfully
agitated. Another remarkable circumstance in his case, was, that

when he was sent in August, 1799, with a large number of other
children, from Saint Pancras workhouse, to a cotton-mill near
Nottingham, he bore amongst his comrades, the name of Parson,
and retained it afterwards till he had served considerably longer than
his fourteen years, and then, when his Indentures were at last
relinquished, and not till then, the young man found he had been
apprenticed by the name of Robert Blincoe. I urged the probability,
that his right indenture might, in the change of masters that took
place, or the careless indifference of his last master, have been given
to another boy, and that to the one given to him, bearing the name
of Blincoe, he had no just claim. This reasoning he repelled, by
steadily and consistently asserting, he fully recollected having heard
it said his real name was Blincoe, whilst he remained at Saint
Pancras workhouse. His indentures were dated the 15th August,
1799. If, at this time, he was seven years of age, which is by no
means certain, he was born in 1792, and in 1796, was placed in
Pancras workhouse. With these remarks I close this preliminary
matter, and happy should I be, if the publication of these facts
enables the individual to whom they relate, to remove the veil which
has hitherto deprived him of a knowledge of his parentage, a
privation which he still appears to feel with undiminished intensity of
grief.
Two years have elapsed, since I first began to take notes of
Blincoe’s extraordinary narrative. At the close of 1822 and beginning
of 1823, I was seized with a serious illness, which wholly prevented
my publishing this and other important communications. The
testimony of a respectable surgeon, who attended me, as any in the
country, even ocular demonstration of my enfeebled state, failed to
convince some of the cotton spinners, that my inability was not
feigned, to answer some sinister end; and such atrocious conduct
was pursued towards me, as would have fully justified a prosecution
for conspiracy. Animated by the most opposite views, the worst of
miscreants united to vilify and oppress me; the one wanting to get
my papers, in order, by destroying them, to prevent the enormities
of the cotton masters being exposed; and another, traducing my

character, and menacing my life, under an impression that I had
basely sold the declarations and communications received from
oppressed work-people to their masters. By some of those
suspicious, misjudging people, Blincoe was led away. He did not,
however, at any time, or under any circumstances, retract or deny
any part of his communications, and, on the 18th and 19th of March,
1824, of his own free will, he not only confirmed all that he had
communicated in the spring of 1822, with many other traits of
suffering, not then recollected, but furnished me with them. It has,
therefore, stood the test of this hurricane, without its authenticity
being in any one part questioned or impaired. The authenticity of
this narrative is, therefore, entitled to greater credit, than much of
the testimony given by the owners of cotton-factories, or by
professional men on their behalf, as will, in the course of this
narrative, be fully demonstrated, by evidence wholly incontrovertible.
If, therefore, it should be proved, that atrocities to the same extent,
exist no longer; still, its publication, as a preventative remedy, is no
less essential to the protection of parish paupers and foundlings. If
the gentlemen of Manchester and its vicinity, who acted in 1816, c.,
in conjunction with the late Mr. Nathaniel Gould, had not made the
selection of witnesses too much in the power of incompetent
persons, Robert Blincoe would have been selected in 1819, as the
most impressive pleader in behalf of destitute and deserted children.

CHAP. II.
Of the few adventures of Robert Blincoe, during his residence in
old Saint Pancras workhouse, the principal occurred when he had
been there about two years. He acknowledges he was well fed,
decently clad, and comfortably lodged, and not at all overdone, as
regarded work; yet, with all these blessings in possession, this
destitute child grew melancholy. He relished none of the humble
comforts he enjoyed. It was liberty he wanted. The busy world lay
outside the workhouse gates, and those he was seldom, if ever
permitted to pass. He was cooped up in a gloomy, though liberal sort
of a prison-house. His buoyant spirits longed to rove at large. He
was too young to understand the necessity of the restraint to which
he was subjected, and too opinionative to admit it could be intended
for his good. Of the world he knew nothing, and the society of a
workhouse was not very well calculated to delight the mind of a
volatile child. He saw givers, destitute of charity, receivers of insult,
instead of gratitude, witnessed little besides sullenness and
discontent, and heard little but murmurs or malicious and slanderous
whispers. The aged were commonly petulant and miserable—the
young demoralized and wholly destitute of gaiety of heart. From the
top to the bottom, the whole of this motley mass was tainted with
dissimulation, and he saw the most abhorrent hypocrisy in constant
operation. Like a bird newly caged, that flutters from side to side,
and foolishly beats its wings against its prison walls, in hope of
obtaining its liberty, so young Blincoe, weary of confinement and
resolved, if possible to be free, often watched the outer gates of the
house, in the hope, that some favourable opportunity might facilitate
his escape. He wistfully measured the height of the wall, and found
it too lofty for him to scale, and too well guarded were the gates to
admit of his egress unnoticed. His spirits, he says, which were
naturally lively and buoyant, sank under this vehement longing after

liberty. His appetite declined, and he wholly forsook his usual sports
and comrades. It is hard to say how this disease of the mind might
have terminated, if an accident had not occurred, which afforded a
chance of emerging from the lifeless monotony of a workhouse, and
of launching into the busy world, with which he longed to mingle.
Blincoe declares, he was so weary of confinement, he would gladly
have exchanged situations with the poorest of the poor children,
whom, from the upper windows of the workhouse, he had seen
begging from door to door, or, as a subterfuge, offering matches for
sale. Even the melancholy note of the sweep-boy, whom, long
before day, and in the depths of winter, in frost, in snow, in rain, in
sleet, he heard pacing behind his surly master, had no terrors for
him. So far from it, he envied him his fortune, and, in the fulness of
discontent, thought his own state incomparably more wretched. The
poor child was suffering under a diseased imagination, from which
men of mature years and elaborate culture are not always free. It
filled his heart with perverted feelings—it rendered the little urchin
morose and unthankful, and, as undeserving of as he was insensible
to, the important benefits extended to him by a humane institution,
when helpless, destitute and forlorn.
From this state of early misanthropy, young Blincoe was suddenly
diverted, by a rumour, that filled many a heart among his comrades
with terror, viz. that a day was appointed, when the master-sweeps
of the metropolis were to come and select such a number of boys as
apprentices, till they attained the age of 21 years, as they might
deign to take into their sable fraternity. These tidings, that struck
damp to the heart of the other boys, sounded like heavenly music to
the ears of young Blincoe:—he anxiously inquired of the nurses if the
news were true, and if so, what chance there was of his being one
of the elect. The ancient matrons, amazed at the boy’s temerity and
folly, told him how bitterly he would rue the day that should consign
him to that wretched employment, and bade him pray earnestly to
God to protect him from such a destiny. The young adventurer heard
these opinions with silent contempt. Finding, on farther inquiry, that
the rumour was well founded, he applied to several menials in the

house, whom he thought likely to promote his suit, entreating them
to forward his election with all the interest they could command!
Although at this time he was a fine grown boy, being fearful he
might be deemed too low in stature, he accustomed himself to walk
in an erect posture, and went almost a tip-toe;—by a ludicrous
conceit, he used to hang by the hands to the rafters and
balustrades, supposing that an exercise, which could only lengthen
his arms, would produce the same effect on his legs and body. In
this course of training for the contingent honour of being chosen by
the master-sweeps, as one fit for their use,—with a perseverance
truly admirable, his tender age considered, young Blincoe continued
till the important day arrived. The boys were brought forth, many of
them in tears, and all except Blincoe, very sorrowful. Amongst them,
by an act unauthorised by his guardians, young Blincoe contrived to
intrude his person. His deportment formed a striking contrast to that
of all his comrades; his seemed unusually high: he smiled as the
grim looking fellows approached him; held his head as high as he
could, and, by every little artifice in his power, strove to attract their
notice, and obtain the honour of their preference. While this
fatherless and motherless child, with an intrepid step, and firm
countenance, thus courted the smiles of the sooty tribe, the rest of
the boys conducted themselves as if they nothing so much dreaded,
as to become the objects of their choice, and shrunk back from their
touch as if they had been tainted by the most deadly contagion. Boy
after boy was taken, in preference to Blincoe, who was often
handled, examined, and rejected. At the close of the show, the
number required was elected, and Blincoe was not among them! He
declared, that his chagrin was inexpressible, when his failure was
apparent.
Some of the sweeps complimented him for his spirit, and, to
console him, said, if he made a good use of his time, and contrived
to grow a head taller, he might do very well for a fag, at the end of a
couple of years. This disappointment gave a severe blow to the
aspiring ambition of young Blincoe, whose love of liberty was so
ardent, that he cared little about the sufferings by which, if attained,

it was likely to be alloyed. The boys that were chosen, were not
immediately taken away. Mingling with these, some of them said to
our hero, the tears standing in their eyes:—“why, Parson, can you
endure the thoughts of going to be a chimney-sweep? I wish they
would take you instead of me.” “So do I, with all my heart,” said
Blincoe, “for I would rather be any where than here.” At night, as
Blincoe lay tossing about, unable to sleep, because he had been
rejected, his unhappy associates were weeping and wailing, because
they had been accepted! Yet, his heart was not so cold as to be
unaffected by the wailings of those poor children, who, mournfully
anticipating the horrors of their new calling, deplored their
misfortune in the most touching terms. They called upon their
parents, who, living or dead, were alike unable to hear them, to
come and save them! What a difference of feeling amongst children
of the same unfortunate class! The confinement that was so
wearisome to young Blincoe, must have been equally irksome to
some of his young associates; therefore, the love of liberty could not
have been its sole cause,—there was another and a stronger reason
—all his comrades had friends, parents, or relations: poor Blincoe
stood alone! no ties of consanguinity or kindred bound him to any
particular portion of society, or to any place—he had no friend to
soothe his troubled mind—no domestic circle to which, though
excluded for a time, he might hope to be reunited. As he stood thus
estranged from the common ties of nature, it is the less to be
wondered at, that, propelled by a violent inclination to a rambling
life, and loathing the restraint imposed by his then condition, he
should indulge so preposterous a notion, as to prefer the wretched
state of a sweeping-boy. Speaking on this subject, Blincoe said to
me, “If I could penetrate the source of my exemption from the
sorrow and consternation so forcibly expressed by my companions, it
would probably have been resolved by the peculiarity of my destiny,
and the privation of those endearing ties and ligatures which cement
family circles. When the friends, relatives, parents of other children
came to visit them, the caresses that were sometimes exchanged,
the joy that beamed on the faces of those so favoured, went as
daggers to my heart; not that I cherished a feeling of envy at their

good fortune; but that it taught me more keenly to feel my own
forlorn condition. Sensations, thus, excited, clouded every festive
hour, and, young as I was, the voice of nature, instinct, if you will,
forced me to consider myself as a moral outcast, as a scathed and
blighted tree, in the midst of a verdant lawn.”
I dare not aver, that such were the very words Blincoe used, but
they faithfully convey the spirit and tendency of his language and
sentiments. Blincoe is by no means deficient in understanding: he
can be witty, satirical, and pathetic, by turns, and he never showed
himself to such advantage, as when expatiating upon the desolate
state to which his utter ignorance of his parentage had reduced him.
During Blincoe’s abode at St. Pancras, he was inoculated at the
Small Pox Hospital. He retained a vivid remembrance of the copious
doses of salts he had to swallow, and that his heart heaved, and his
hand shook as the nauseous potion approached his lips. The old
nurse seemed to consider such conduct as being wholly unbecoming
a pauper child; and chiding young Blincoe, told him, he ought to
“lick his lips,” and say thank you, for the good and wholesome
medicine provided for him at the public expense; at the same time,
very coarsely reminding him of the care that was taken to save him
from an untimely death by catching the small-pox in the natural way.
In the midst of his subsequent afflictions, in Litton Mill, Blincoe,
declared, he often lamented having, by this inoculation, lost a
chance of escaping by an early death, the horrible destiny for which
he was preserved.
From the period of Blincoe’s disappointment, in being rejected by
the sweeps, a sudden calm seems to have succeeded, which lasted
till a rumour ran through the house, that a treaty was on foot
between the Churchwardens and Overseers of St. Pancras, and the
owner of a great cotton factory, in the vicinity of Nottingham, for the
disposal of a large number of children, as apprentices, till they
become twenty-one years of age. This occurred about a
twelvemonth after his chimney-sweep miscarriage. The rumour itself
inspired Blincoe with new life and spirits; he was in a manner

intoxicated with joy, when he found, it was not only confirmed, but
that the number required was so considerable, that it would take off
the greater part of the children in the house,—poor infatuated boy!
delighted with the hope of obtaining a greater degree of liberty than
he was allowed in the workhouse,—he dreamed not of the misery
that impended, in the midst of which he could look back to Pancras
as to an Elysium, and bitterly reproach himself for his ingratitude
and folly.
Prior to the show-day of the pauper children to the purveyor or
cotton master, the most illusive and artfully contrived falsehoods
were spread, to fill the minds of those poor infants with the most
absurd and ridiculous errors, as to the real nature of the servitude,
to which they were to be consigned. It was gravely stated to them,
according to Blincoe’s statement, made in the most positive and
solemn manner, that they were all, when they arrived at the cotton-
mill, to be transformed into ladies and gentlemen: that they would
be fed on roast beef and plum-pudding—be allowed to ride their
masters’ horses, and have silver watches, and plenty of cash in their
pockets. Nor was it the nurses, or other inferior persons of the
workhouse, with whom this vile deception originated; but with the
parish officers themselves. From the statement of the victims of
cotton-mill bondage, it seems to have been a constant rule, with
those who had the disposal of parish children, prior to sending them
off to cotton-mills, to fill their minds with the same delusion. Their
hopes being thus excited, and their imaginations inflamed, it was
next stated, amongst the innocent victims of fraud and deception,
that no one could be compelled to go, nor any but volunteers
accepted.
When it was supposed at St. Pancras, that these excitements had
operated sufficiently powerful to induce a ready acquiescence in the
proposed migration, all the children, male and female, who were
seven years old, or considered to be of that age, were assembled in
the committee-room, for the purpose of being publicly examined,
touching their health, and capacity, and what is almost incredible
touching their willingness to go and serve as apprentices, in the way

and manner required! There is something so detestable, in this
proceeding, that any one might conclude, that Blincoe had been
misled in his recollections of the particulars; but so many other
sufferers have corroborated his statement, that I can entertain no
doubt of the fact. This exhibition took place in August 1799, and
eighty boys and girls as parish apprentices, and till they had
respectively acquired the age of twenty-one years, were made over
by the churchwardens and overseers of Saint Pancras parish, to
Messrs. Lamberts’, cotton-spinners, hosiers and lace-men, of St.
Mary’s parish, Nottingham, the owners of Lowdam Mill. The boys,
during the latter part of their time, were to be instructed in the trade
of stocking weaving—the girls in lace-making. There was no
specification whatever, as to the time their masters were to be
allowed to work these poor children, although, at this period, the
most abhorrent cruelties were notoriously known to be exercised, by
the owners of cotton-mills, upon parish apprentices. According to
Blincoe’s testimony, so powerfully had the illusions, purposely spread
to entrap these poor children, operated, and so completely were
their feeble minds excited, by the blandishments held out to them,
that they almost lost their wits. They thought and talked of nothing
but the scenes of luxury and grandeur, in which they were to move.
Nor will the reflecting reader feel surprised at this credulity, however
gross, when he considers the poor infants imagined there were no
greater personages than the superiors, to whom they were, as
paupers, subjected, and that, it was those identical persons, by
whom their weak and feeble intellects had thus been imposed upon.
Blincoe describes his conduct to have been marked by peculiar
extravagance. Such was his impatience, he could scarcely eat or
sleep, so anxiously did he wait the hour of emancipation. The poor
deluded young creatures were so inflated with pride and vanity, that
they strutted about like so many dwarfish and silly kings and
queens, in a mock tragedy. “We began” said Blincoe “to treat our old
nurses with airs of insolence and disdain—refused to associate with
children, who, from sickness, or being under age, had not been
accepted; they were commanded to keep their distance; told to
know their betters; forbidden to mingle in our exalted circle! Our

little coterie was a complete epitome of the effects of prosperity in
the great world. No sooner were our hearts cheered by a prospect of
good fortune, than its influence produced the sad effects recited.
The germ of those hateful vices, arrogance, selfishness and
ingratitude, began to display themselves even before we had tasted
the intoxicating cup. But our illusion soon vanished, and we were
suddenly awakened from the flattering dream, which consigned the
greater part of us to a fate more severe than that of the West Indian
slaves, who have the good fortune to serve humane owners.” Such
were Blincoe’s reflections in May 1822.
It appears that the interval was not long, which filled up the space
between their examination, acceptance, and departure from St.
Pancras workhouse, upon their way to Nottingham; but short as it
was, it left room for dissension. The boys could not agree who
should have the first ride on their masters’ horses, and violent
disputes arose amongst the girls, on subjects equally ludicrous. It
was afterwards whispered at Lowdam Mill, that the elder girls,
previous to leaving Pancras, began to feel scruples, whether their
dignity would allow them to drop the usual bob-curtsey to the
master or matron of the house, or to the governess by whom they
had been instructed to read, or work by the needle. Supposing all
these follies to have been displayed to the very letter, the poor
children were still objects of pity; the guilt rests upon those by
whom they had been so wickedly deceived!
Happy, no doubt, in the thought of transferring the burthen of the
future support of fourscore young paupers to other parishes, the
churchwardens and overseers distinguished the departure of this
juvenile colony by acts of munificence. The children were completely
new clothed, and each had two suits, one for their working, the
other for their holiday dress—a shilling in money, was given to each
—a new pocket handkerchief—and a large piece of gingerbread. As
Blincoe had no relative of whom to take leave, all his anxiety was to
get outside the door. According to his own account, he was the first
at the gate, one of the foremost who mounted the waggon, and the
loudest in his cheering. In how far the parents or relatives of the rest

of the children consented to this migration; if they were at all
consulted, or even apprised of its being in contemplation, formed no
part of Blincoe’s communications. All he stated was, that the whole
of the party seemed to start in very high spirits. As to his own
personal conduct, Blincoe asserts, he strutted along dressed in
party-coloured parish clothing, on his way to the waggon, no less
filled with vanity than with delusion: he imagined he was free, when
he was in fact legally converted into a slave; he exulted in the
imaginary possession of personal liberty, when he was in reality a
prisoner. The whole convoy were well guarded by the parish beadles
on their way to the waggons; but those officers, bearing their
staves, the children were taught to consider as a guard of honour. In
addition to the beadles, there was an active young man or two,
appointed to look after the passengers of the two large waggons, in
their conveyance to Nottingham. Those vehicles, and very properly
too, were so secured, that when once the grated doors were locked,
no one could escape. Plenty of clean straw was strewed in the beds,
and no sooner were the young fry safely lodged within, than they
began throwing it over one another and seemed delighted with the
commencement of their journey. A few hours progress considerably
damped this exultation. The inequality of the road, and the heavy
jolts of the waggon, occasioned them many a bruise. Although it was
the middle of August, the children felt very uncomfortable. The
motion of the heavy clumsey vehicle, and so many children cooped
up in so small a space, produced nausea and other results, such as
sometimes occur in Margate boys. Of the country they passed
through, the young travellers saw very little.—Blincoe thinks the
children were suffered to come out of the waggon to walk through
St. Alban’s. After having passed one night in the waggon, many of
the children began to repent, and express a wish to return. They
were told to have patience, till they arrived at Messrs. Lamberts,
when, no doubt, those gentlemen would pay every attention to their
wishes, and send back to St. Pancras, those who might wish to
return. Blincoe, as might have been expected, was not one of those
back-sliders—he remained steady to his purpose, exulting in the
thought, that every step he advanced brought him nearer to the

desired spot, where so many enviable enjoyments awaited him, and
conveyed him farther and farther from the detested workhouse!
Blincoe being so overjoyed with the fine expectations he was to
receive at Lowdam Mill, he spent his shilling at Leicester in apples.
The greater part of the children were much exhausted, and not a
few of them seriously indisposed, before they arrived at Nottingham.
When the waggons drew up near the dwelling and warehouse of
their future master, a crowd collected to see the live stock that was
just imported from the metropolis, who were pitied, admired, and
compared to lambs, led by butchers to slaughter! Care was taken
that they should not hear or understand much of this sort of
discourse. The boys and girls were distributed, some in the kitchen,
others in a large ware-room, washed, combed and supplied with
refreshments; but there were no plum-pudding—no roast beef, no
talk of the horses they were to ride, nor of the watches and fine
clothing that they had been promised. Many looked very mournful;
they had been four days travelling to Nottingham: at a more
advanced period of their lives, a travel to the East Indies might not
have been estimated as a much more important or hazardous
undertaking. After having been well refreshed, the whole of the boys
and girls were drawn up in rows, to be reviewed by their masters,
their friends and neighbours. In Blincoe’s estimation, their masters,
Messrs. Lamberts’, were “stately sort of men.” They looked over the
children and finding them all right, according to the invoice, exhorted
them to behave with proper humility and decorum. To pay the most
prompt and submissive respects to the orders of those who would
be appointed to instruct and superintend them at Lowdam Mill, and
to be diligent and careful, each one to execute his or her task, and
thereby avoid the punishment and disgrace which awaited idleness,
insolence, or disobedience. This harangue, which was delivered in a
severe and dictatorial tone, increased their apprehensions, but not
one durst open a mouth to complain. The masters and their servants
talked of the various sorts of labour to which the children were to
apply themselves, and to the consternation and dismay of Blincoe
and his associates, not the least allusion was made to the many fine

things which had so positively been promised them whilst in London.
The conversation which Blincoe heard, seemed to look forward to
close, if not to unremitting toil, and the poor boy had been filled with
expectations, that he was to work only when it pleased him; to have
abundance of money and fine clothes—a watch in his pocket, to
feast on roast beef and plum-pudding, and to ride his masters
horses. His hopes, however were, not wholly extinguished, because
Nottingham was not Lowdam Mill, but his confidence was greatly
reduced, and his tone of exultation much lowered.
The children rested one night at Nottingham in the warehouses of
their new masters—the next day they were led out to see the castle,
Mortimer-hole and other local curiosities, in the forest of Sherwood,
which are so celebrated by bards of ancient times. Many shoes,
bonnets, and many other articles of clothing having been lost upon
the journey, others were supplied—but withal Blincoe found himself
treated as a parish orphan, and he calculated on being received and
treated as if he had been a gentleman’s son sent on a visit to the
house of a friend or relative. By the concurring testimony of other
persons who had been entrapped by similar artifices, it appears
certain, that the purveyors of infant labourers to supply the masters
of cotton and silk factories with cheap labourers, adopted this vile,
unmanly expedient, in most of their transactions. It will be seen, by
the evidence of Sir Robert Peel, Baronet, David Owen, Esq. and
other witnesses examined in 1816, that, when children were first
wanted to attend machinery in cotton-factories, such was the
aversion of parents and guardians to this noxious employment, that
scarcely any would submit to consign their offspring to those mills,
the owners of which, under the specious pretext of diminishing the
burdens occasioned by poor-rates, prevailed on churchwardens and
overseers, to put their infant paupers into their hands. Since then,
by a gradual progress of poverty and depravity, in the county of
Lancashire alone, there are some thousand fathers, mothers, and
relatives, who live upon the produce of infant labour, though alloyed
by the dreadful certainty, that their gain is acquired by the sacrifice
of their children’s health and morals, and too frequently of their

lives, whereby the fable of Saturn devouring his children, seems
realised in modern times.

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