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Springer Monographs in Mathematics

For further volumes published in this series,
www.springer.com/series/3733


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R.M. Dudley • R. Norvaiša

Concrete Functional Calculus


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R.M. Dudley
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA 02139-4307
USA


R. Norvaiša
Institute of Mathematics and Informatics
LT-08663 Vilnius

Lithuania


ISSN 1439-7382
ISBN 978-1-4419-6949-1
e-ISBN 978-1-4419-6950-7
DOI 10.1007/978-1-4419-6950-7
Springer New York Dordrecht Heidelberg London
Mathematics Subject Classification (2010): 28Bxx, 45Gxx, 46Txx, 60Gxx

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Preface

The reader of this book will need some background in real analysis, as
in the first half of the first author’s book Real Analysis and Probability. The
book should be accessible to graduate students with that background as well

as researchers. An interest in probability will also help with motivation.
Although on some topics we may have presented more or less final results,
there are others that leave openings for research. The impetus for much of
the work came from others’ work in mathematical statistics, but we do not
include any applications to statistics.
The book is mainly about some aspects of nonlinear analysis, some not
much studied and some others previously studied but not in the same ways,
and their applications to probability, as in the final Chapter 12. More specifically (to explain the book’s title) we consider existence and smoothness questions for some concrete nonlinear operators acting on some concrete Banach
spaces of functions. The book has relatively small overlaps, of the order of one
or two chapters, with any previous book except for two lecture note volumes
by the authors.
Here is a first example of what is done and distinctive in this book. If F
and G are two functions such that F is defined on the range of G, one can
form the composition H(x) ≡ F (G(x)) ≡ (F ◦ G)(x). When one mentions
differentiability and composition, mathematicians tend to think of the chain
rule, which is indeed an important fact, but we consider differentiability of
the two-function composition operator we call T C which takes the pair of
functions (F, G) into the function H. To take the derivative of this operator,
we will assume F and G take values in Banach spaces. The domain of G
need not not have a linear or topological structure (it may be a measure
space). What the differentiation will mean at some F, G is to take functions f
and g approaching 0 in corresponding spaces, and to represent the increment
(F + f ) ◦ (G + g) − F ◦ G as A(f ) + B(g) plus a remainder, where A and
B are linear operators and the remainder becomes small in norm relative to
f, g → 0. The operator T C is linear in F for fixed G, but the remainder
contains a term f ◦ (G + g) − f ◦ G which still depends on both f and g.


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VI


Preface

It seems to us that T C is a very natural operator deserving attention. It is
treated in Chapter 8. Relatively more attention, e.g. Appell and Zabrejko [3],
has been given to the operator G → F ◦ G for fixed (nonlinear) F , and its
extension to the case where F is a function of two variables, say x and y, and
one forms a new function H(x) = F (x, G(x)). Such operators are treated in
Chapters 6 and 7.
Among the most familiar of all Banach spaces are the Lp spaces, of equivalence classes of functions f such that |f |p is integrable for a given measure µ,
where 1 ≤ p < ∞. A question is: for given p and s in [1, ∞), for what space(s)
F of functions from R into R do we have Fr´echet differentiability (defined in
Chapter 5) of T C from F × Ls into Lp , say where the measure space is [0, 1]
with Lebesgue measure? It turns out that for 1 ≤ s < p, we get degeneracy:
even for fixed F , it must be constant (Corollary 7.36). For 1 ≤ p < s < ∞,
consider the following. Let f be a real-valued function on R and 1 ≤ p < ∞.
n
p
Let vp (f ) be the supremum of
j=1 |f (xj ) − f (xj−1 )| over all partitions
x0 = a < x1 < · · · < xn = b and all n, called the p-variation of f . Finiteness
of this is ordinary bounded variation when p = 1. Let F = Wp (R) be the space
of all f such that vp (f ) < ∞, with the norm f [p] = vp (f )1/p + f sup where
f sup := supx |f (x)|. Theorem 8.9 shows that T C is Fr´echet differentiable
from Wp (R) × Ls into Lp of a finite measure space (Ω, S, µ) at suitable F
and G. Namely, F is differentiable, and its derivative F ′ satisfies a further
condition. The image measure µ ◦ G−1 has a bounded density with respect to
Lebesgue measure. The theorem gives a bound on the remainder in the differentiation of a given order in terms of s and p. The two paragraphs preceding
Theorem 8.9 indicate how some of the conditions assumed are necessary or
best possible and in particular, optimality of the Wp norm on f for bounding

the remainder term f ◦ (G + g) − f ◦ G (Proposition 7.28).
Countably additive signed measures are familiar objects in real analysis.
For purposes of this book we need to consider Stieltjes-type integrals f dg
where neither the integrator g nor the integrand f is of bounded variation.
We found it useful to consider as integrators interval functions defined as follows. An interval function µ is a function such that µ(A) is defined for all
intervals A, which may be restricted to subintervals of some given interval.
Then µ is called additive if µ(A ∪ B) = µ(A) + µ(B) for any two disjoint
intervals A, B such that A ∪ B is an interval, and µ is called upper continuous
if µ(An ) → µ(A) whenever intervals An decrease down to A. These properties would follow from, but do not imply, existence of an extension of µ to
a countably additive signed measure. For example, if F is a right-continuous
function with left limits, not necessarily of bounded variation, one can define
µF ((c, d]) := F (d) − F (c) and define µF for other intervals by taking limits, giving an additive, upper continuous interval function µF . For µ upper
continuous we define vp (µ), for µ which may be Banach-valued, as the suprep
mum of
over all finite disjoint collections of intervals Ai . For
i µ(Ai )
not necessarily additive interval functions, the p-variation needs to be defined
differently.


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Preface

VII

Beside composition, another natural operator is the product integral,
which takes the coefficient functions (such as C(t) in (1.10) below) of a system of linear ordinary differential equations into a solution, under suitable
conditions. Despite linearity of the system, the operator is nonlinear in the
coefficients. Some entire books have been written on this operator. The product integral will take values in Banach algebra, namely, a Banach space in
which a multiplication is defined satisfying usual algebraic conditions and

xy ≤ x y . A precise definition and some of the theory of Banach algebras are given in Chapter 4. An interval function µ with values in a Banach
algebra will be called multiplicative if µ(A ∪ B) = µ(B)µ(A) holds for any disjoint intervals A, B such that A∪B is an interval, with s < t for each s ∈ A and
t ∈ B. For 1 ≤ p < 2, a product integral operator µ → (1I + dµ) is defined
from additive to multiplicative interval functions and is an entire analytic
function (as defined in Chapter 5) with respect to p-variation norms (Theorem 9.51) and serves to solve differential and integral equations. Whereas, for
p > 2, finite p-variation of an additive, upper continuous µ does not imply
that the product integral even exists (Theorem 9.11).
Since p-variation gives sharp results about two natural operators, let’s
return to point functions f and consider the space Wp (R) for 1 ≤ p < ∞. If G
is any homeomorphism of R, in other words, a continuous, strictly monotone
(increasing or decreasing) function from R onto itself, the map f → f ◦ G
preserves Wp and its norm. Invariance under this very large group holds for
the spaces of all bounded continuous functions or all bounded functions with
the norm f sup . Other commonly considered spaces such as Sobolev spaces
are of course highly useful, but they are invariant under much more restricted
transformations. Whereas, the supremum norm gives no control of oscillations
of a function and the Wp norms do. We suppose the good properties of · [p]
will give other uses than those we have found.
We also treat integrals, although with relatively little attention to the
Lebesgue integral. Rather, let at first f and g be real functions of a real
b
variable and consider Stieltjes-type integrals (f, g) → a f dg where neither f
nor g is necessarily of bounded variation. The given bilinear functional can
be defined on various domains f ∈ F, g ∈ G as will be seen. If f and g have
suitable infinite-dimensional ranges, the integral can also be extended.
If 1 ≤ p < ∞, 1 ≤ q < ∞, vp (f ) < ∞, vq (g) < ∞, and p−1 + q −1 > 1,
then a Stieltjes-type integral f dg can be defined, as had been shown by E.
R. Love and L. C. Young in the late 1930’s, with a corresponding inequality
we call the Love–Young inequality (Corollary 3.91) and use often. Because of
bilinearity, the differentiability is then immediate and simple.

Chapter 12 on probability and p-variation includes results from several
research papers. Among others, Theorem 12.27 gives bounded p-variation of
the sample paths of Markov processes (with values in metric spaces) for 2 <
p < ∞ under a mild condition on expected lengths of increments, shown
to be sharp. Corollary 12.43 extends the celebrated Koml´
os–Major–Tusn´
ady
theorem on convergence of the classical empirical process to a Brownian bridge


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VIII

Preface

from the supremum norm to p-variation norms for 2 < p < ∞ with slower,
but sharp, rates of convergence. Theorem 12.40 gives a sharp bound on the
growth of the p-variation of the classical empirical process for 1 ≤ p < 2.
These facts were published in three papers in Annals of Probability (one coauthored by one of us). Proposition 12.54 gives an example due to Terry Lyons,
showing that for some processes Xt and Yt which are each Brownian bridges,
1
but which have an unusual joint distribution, an integral 0 Xt dYt cannot be
defined by any of the usual methods. The full work of Lyons and co-workers
on “rough paths” is in progress and is beyond the scope of this book. In some
cases where compact or “Hadamard” differentiability had been proved in the
statistics literature, it might well be replaced by Fr´echet differentiability with
respect to p-variation norms, with a gain of remainder bounds. This and other
problems we here leave as opportunities for readers.
Some parts of the book appeared, in different forms, in our earlier lecture
note volumes Dudley and Norvaiˇsa 1998 [55] and 1999 [54]. Improvements on

or corrections to earlier results of ours or others are incorporated. The book
also includes some new results published here for the first time as far as we
know, as will be mentioned in the text or Notes for each.
Guide to the reader: Starred sections, specifically Sections *2.7 and *3.4,
are not referred to in later chapters. Chapters of the book depend on earlier
chapters as follows: Chapters 1 through 3 are basic in that Chapter 1 is a
rather short introduction, and all later chapters refer to Chapter 3 many
times each and directly or indirectly to Chapter 2. In the further sequence of
chapters 4 through 8, each chapter has many references to the preceding one
and directly or indirectly to intervening chapters. The Appendix relates only
to Chapter 7. Chapter 12 on stochastic processes refers to, beyond Chapter 3,
only two propositions in Chapter 9. Chapter 11, on Fourier series, refers only
to the basic Chapters 1-3. Chapter 10, on nonlinear differential and integral
equations, refers twice to Chapter 9, once to Chapter 7, once to Chapter 5,
and many times to Chapter 3. Chapter 9, on multiplicative interval functions,
the product integral, and linear differential and integral equations, refers to
Chapters 1 through 6.
Acknowledgments. We thank B. M. Garay, Richard Gill, and Timothy
Nguyen for advice and help with some points in the book.
MIT, Cambridge, Massachusetts
Institute of Mathematics and Informatics, Vilnius
February 2010
1

2

Richard M. Dudley1
Rimas Norvaiˇsa2

R. M. Dudley was partially supported by U. S. National Science Foundation

Grants.
R. Norvaiˇsa was partially supported by U. S. National Science Foundation Grants,
by Poduska Family Foundation grants and by Lithuanian State Science and Studies Foundation grants Nos. T-21/07, T-16/08, and T-68/09.


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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
1

Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 How to Define ∫ab f dg? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Some Integral and Differential Equations . . . . . . . . . . . . 7
1.3 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Notation and Elementary Notions . . . . . . . . . . . . . . . . . . . 10

2

Definitions and Basic Properties of Extended
Riemann–Stieltjes Integrals . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Regulated and Interval Functions . . . . . . . . . . . . . . . . . . .
2.2 Riemann–Stieltjes Integrals . . . . . . . . . . . . . . . . . . . . . . . .
2.3 The Refinement Young–Stieltjes and Kolmogorov
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Relations between RS, RRS, and RYS Integrals . . . . . .
2.5 The Central Young Integral . . . . . . . . . . . . . . . . . . . . . . . .
2.6 The Henstock–Kurzweil Integral . . . . . . . . . . . . . . . . . . . .
*2.7 Ward–Perron–Stieltjes and Henstock–Kurzweil Integrals

2.8 Properties of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Relations between Integrals . . . . . . . . . . . . . . . . . . . . . . . .
2.10 Banach-Valued Contour Integrals and Cauchy Formulas
2.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

17
17
24
29
46
51
56
62
71
87
90
98

Φ-variation and p-variation; Inequalities for Integrals 103
3.1 Φ-variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2 Interval Functions, Φ-variation, and p-variation . . . . . . . 116
3.3 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
*3.4 A Necessary and Sufficient Condition for Integral
Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.5 Sufficient Conditions for Integrability . . . . . . . . . . . . . . . . 168
3.6 Love–Young Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 173



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3.7 Existence of Extended Riemann–Stieltjes Integrals . . . . 191
3.8 Convolution and Related Integral Transforms . . . . . . . . . 196
3.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4

Banach Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
4.1 Ideals and Normed Algebras . . . . . . . . . . . . . . . . . . . . . . . 216
4.2 The Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.3 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
4.4 Holomorphic Functions of Banach Algebra Elements . . 225
4.5 Complexification of Real Banach Algebras . . . . . . . . . . . 231
4.6 A Substitution Rule for the Kolmogorov Integral . . . . . 233
4.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

5

Derivatives and Analyticity in Normed Spaces . . . . . . 237
5.1 Polynomials and Power Series . . . . . . . . . . . . . . . . . . . . . . 239
5.2 Higher Order Derivatives and Taylor Series . . . . . . . . . . 251
5.3 Taylor’s Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.4 Tensor Products of Banach Spaces . . . . . . . . . . . . . . . . . . 269
5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

6


Nemytskii Operators on Some Function Spaces . . . . . 273
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
6.2 Remainders in Differentiability . . . . . . . . . . . . . . . . . . . . . 274
6.3 Higher Order Differentiability and Analyticity . . . . . . . . 276
6.4 Autonomous Nemytskii Operators on Wp Spaces . . . . . . 293
6.5 Nemytskii Operators on Wp Spaces . . . . . . . . . . . . . . . . . 300
6.6 Higher Order Differentiability and Analyticity on Wp
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

7

Nemytskii Operators on Lp Spaces . . . . . . . . . . . . . . . . . . 335
7.1 Acting, Boundedness, and Continuity Conditions . . . . . 335
7.2 Hă
older Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
7.3 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
7.4 Higher Order Differentiability and Finite Taylor Series . 368
7.5 Examples where NF Is Differentiable and F Is Not . . . . 378
7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

8

Two-Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . 393
8.1 Overview; General Remarks . . . . . . . . . . . . . . . . . . . . . . . . 393
8.2 Differentiability of Two-Function Composition in
General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
8.3 Measure Space Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8.4 Spaces with Norms Stronger than Supremum Norms . . 401
8.5 Two-Function Composition on Wp Spaces . . . . . . . . . . . . 404



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XI

8.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
9

Product Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
9.1 Multiplicative Interval Functions and Φ-Variation . . . . . 408
9.2 Product Integrals for Real-Valued Interval Functions . . 415
9.3 Nonexistence for p > 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
9.4 Inequalities for Finite Products . . . . . . . . . . . . . . . . . . . . . 425
9.5 The Product Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
9.6 The Strict Product Integral . . . . . . . . . . . . . . . . . . . . . . . 432
9.7 Commutative Banach Algebras . . . . . . . . . . . . . . . . . . . . . 438
9.8 Integrals with Two Integrands . . . . . . . . . . . . . . . . . . . . . . 442
9.9 Duhamel’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
9.10 Smoothness of the Product Integral Operator . . . . . . . . 457
9.11 Linear Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 474
9.12 Integral Equations for Banach-Space-Valued Functions 494
9.13 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499

10

Nonlinear Differential and Integral Equations . . . . . . . 505
10.1 Classical Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . 505
10.2 Picard’s Method in p-Variation Norm, 1 ≤ p < 2 . . . . . . 506

10.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 513
10.4 Boundedness and Convergence of Picard Iterates . . . . . . 527
10.5 Continuity of the Solution Mappings . . . . . . . . . . . . . . . . 536
10.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549

11

Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551
11.1 On the Order of Decrease of Fourier Coefficients . . . . . . 551
11.2 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
11.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568

12

Stochastic Processes and Φ-Variation . . . . . . . . . . . . . . . 571
12.1 Processes with Regulated Sample Functions . . . . . . . . . 571
12.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
12.3 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576
12.4 Gaussian Stochastic Processes . . . . . . . . . . . . . . . . . . . . . 579
12.5 Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
12.6 L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
12.7 Empirical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
12.8 Differentiability of Operators on Processes . . . . . . . . . . 628
12.9 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
12.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 640

Appendix Nonatomic Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 645
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649



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XII

Contents

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665
Index of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669


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1
Introduction and Overview

In this book we are mainly concerned with derivatives of certain specific nonlinear operators on functions. One operator is composition, T C : (f, g) → f ◦g,
where (f ◦g)(x) ≡ f (g(x)). This operator is linear in f for fixed g, but not in g
even for fixed nonlinear f . We call this operator the two-function composition
operator, or T C for short, to distinguish it from the more-studied operator
g → f ◦ g for fixed f , which we call the (autonomous) Nemytskii operator
Nf . The chain rule, on differentiating x → f (g(x)), where x, f , and g may
all have values in Banach spaces, is a very important fact, but it is not directly about either T C or Nf . To differentiate T C will mean to approximate
(f + h) ◦ (g + k) − f ◦ g, asymptotically as h and k approach 0 in suitable
senses, by a sum of linear operators of the functions h and k. The operator
T C will be treated in Chapter 8 and the Nemytskii operator in Chapters 6
and 7.
We will also consider solutions of certain ordinary differential equations
and integral equations, for functions possibly having values in Banach spaces,
and representing such solutions by way of nonlinear operators. A first introduction is given in Section 1.2. A basic nonlinear operator giving solutions of
linear equations, the product integral, is developed in Chapter 9 and applied

to solving equations in Sections 9.11 and 9.12. Chapter 10 treats nonlinear
integral equations for possibly discontinuous functions.
A basic operator on functions on an interval [a, b] is the bilinear Riemann–
b
Stieltjes integral operator (f, g) → a f dg, to be treated in the following
section and then more fully in Chapters 2 and 3. Some general facts about
Banach algebras are reviewed in Chapter 4. Chapter 5 treats differentiability
in general Banach spaces.
The word “concrete” in the title is meant to convey that we consider not
only specific operators as mentioned but also specific function spaces, most
notably p-variation spaces, as will be mentioned in Theorem 1.4 and (1.20),
and frequently throughout most of the book.

R.M. Dudley and R. Norvaiša, Concrete Functional Calculus, Springer Monographs
in Mathematics, DOI 10.1007/978-1-4419-6950-7_1,
© Springer Science+Business Media, LLC 2011

1


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2

1 Introduction and Overview

1.1 How to Define ∫ab f dg?
Two classical ways of defining integrals ∫ab f dg for real-valued functions f and
g on an interval [a, b] with −∞ < a < b < +∞ are the Riemann–Stieltjes
and Lebesgue–Stieltjes integrals. But, as will be seen later in this section and
in Chapter 2, there are multiple ways of defining such integrals. Relations

between them are given in Section 2.9. Different forms of integral can be
useful for different purposes.
The Riemann–Stieltjes integral is defined as follows. A finite sequence
κ = {ti }ni=0 for a positive integer n is called a partition of [a, b] if a = t0 <
t1 < · · · < tn = b. If {ti }ni=0 is a partition of [a, b] then an ordered pair
({ti }ni=0 , {si }ni=1 ) is called a tagged partition of [a, b] if si ∈ [ti−1 , ti ] for i =
1, . . . , n. For a tagged partition τ = ({ti }ni=0 , {si }ni=1 ), |τ | := max1≤i≤n (ti −
ti−1 ) is called the mesh of τ . Let f and g be real-valued functions on [a, b].
For a tagged partition τ = ({ti }ni=0 , {si }ni=1 ) of [a, b], the sum
n

SRS (f, dg; τ ) :=
i=1

f (si ) g(ti ) − g(ti−1 )

is called the Riemann–Stieltjes sum based on τ . The Riemann–Stieltjes integral of f with respect to g is defined and equals C ∈ R if for each ǫ > 0 there
exists a δ > 0 such that |C − SRS (f, dg; τ )| < ǫ for each tagged partition τ of
[a, b] with mesh |τ | < δ. Then we let
b

(RS)

f dg := C = lim SRS (f, dg; τ ).
|τ |→0

a

Closely related to the Riemann–Stieltjes integral is the integral obtained by
replacing the limit as the mesh approaches 0 by the limit in the sense of

refinements of partitions, as follows. A partition κ is a refinement of a partition
λ if λ ⊂ κ as sets. Similarly, a tagged partition τ = (κ, ξ) is a tagged refinement
of a partition λ if κ is a refinement of λ. Now for f and g as before, the
refinement Riemann–Stieltjes integral of f with respect to g is defined and
equals C ∈ R if for each ǫ > 0 there exists a partition λ of [a, b] such that
|C −SRS (f, dg; τ )| < ǫ for each tagged partition τ which is a tagged refinement
of λ. Then we let
b

(RRS)

f dg := lim SRS (f, dg; τ ) := C.
a

τ

If the Riemann–Stieltjes integral exists then so does the refinement Riemann–
Stieltjes integral with the same value, but not conversely (see Proposition 2.13
below and the example following it).
The following concept will be used to formulate sufficient conditions for
existence of integrals. Let f be any real-valued function on an interval [a, b]


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1.1 How to Define ∫ab f dg?

3

with −∞ < a ≤ b < +∞ and let 0 < p < ∞. If a < b, for a partition
κ = {ti }ni=0 of [a, b], the p-variation sum for f over κ will be defined by

n

p

f (ti ) − f (ti−1 ) .

sp (f ; κ) :=
i=1

(1.1)

The p-variation of f on [a, b] is defined as vp (f ; [a, b]) := 0 if a = b and
vp (f ) := vp (f ; [a, b]) := sup sp (f ; κ)

(1.2)

κ

if a < b, where the supremum is over all partitions κ of [a, b]. Then f is said to
be of bounded p-variation on [a, b], or f ∈ Wp [a, b], if and only if vp (f ) < +∞.
Let Vp (f ) := Vp (f ; [a, b]) := vp (f ; [a, b])1/p . A function of bounded p-variation
is clearly bounded. For any bounded real-valued function f on a set S let
f sup := f S,sup := supx∈S |f (x)|. For 1 ≤ p < ∞ let
f

(p)

:=

f


[a,b],(p)

:= Vp (f ),

f

[p]

:=

f

(p)

+ f

sup .

(1.3)

It is easily seen that · (p) is a seminorm, and so · [p] is a norm on Wp [a, b],
using the Hăolder and Minkowski inequalities. Recall the Hă
older inequality for
nite sums: if p1 + q −1 ≥ 1 and 1 ≤ p, q < ∞, then for any nonnegative
numbers {ai , bi }ni=1 ,
n
i=1

n


ai b i ≤

api

1/p

n

bqi

1/q

.

(1.4)

i=1

i=1

This fact is well known for p−1 + q −1 = 1 and follows for p−1 + q −1 ≥ 1
n
since ( i=1 api )1/p is a nonincreasing function of p. Recall also the Minkowski
inequality for finite sums: for any 1 ≤ r < ∞ and any nonnegative numbers
{ai , bi }ni=1 ,
n

(ai + bi )r
i=1


1/r

n

≤

ari
i=1

1/r

n

bri

+

1/r

.

(1.5)

i=1

The best known case of p-variation is for p = 1. A function is said to be
of bounded variation iff it is of bounded 1-variation, and its total variation is
its 1-variation.
Let f ∈ W1 [a, b] and let F (t) := v1 (f ; [a, t]) for a ≤ t ≤ b. Then as is well

known and easily checked, F and F −f are nondecreasing with f ≡ F −(F −f ).
Conversely, if G is any nondecreasing function on [a, b], then v1 (G; [a, b]) =
G(b) − G(a). For any two functions g and h, v1 (g + h) ≤ v1 (g) + v1 (h). Thus,
f ∈ W1 [a, b] if and only if f = G − H for two nondecreasing real-valued
functions G and H on [a, b].
From basic measure and integration theory, e.g. [53, Theorem 3.2.6], recall
that there is a 1–1 correspondence between nondecreasing functions g on (a, b],


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4

1 Introduction and Overview

right-continuous on (a, b), and finite countably additive measures µ on the
Borel sets of (a, b], given by µ((c, d]) = g(d) − g(c), a ≤ c ≤ d ≤ b. The same
equations give a 1–1 correspondence between finite countably additive signed
measures µ on the Borel sets of (a, b] and functions g of bounded variation on
(a, b], right-continuous on (a, b). Let µ := µg for the µ corresponding to such
a g. Then the Lebesgue–Stieltjes integral is defined for a ≤ b by
b

(LS)

f dg :=
a

f dµg

(1.6)


(a,b]

whenever the right side is defined as a Lebesgue integral, for example, for any
bounded, Borel measurable function f . For a = b, the integral is evidently 0.
(Some other authors give definitions which differ as to the point a.)
Let h(t) := 1[1,2] (t) for 0 ≤ t ≤ 2. Then the Riemann–Stieltjes integrals
(RS) ∫02 h dh and (RRS) ∫02 h dh are not defined. Clearly (LS) ∫02 h dh is defined
and equals 1. In this case the Riemann–Stieltjes integrals are remarkably weak.
On the other hand, recall that a real-valued function f on an interval [a, b]
is said to satisfy a Hă
older condition of order α, where 0 < α ≤ 1, if for some
K < ∞, |f (t) − f (s)| ≤ K|t − s|α for any s, t ∈ [a, b]. It is easily seen that a
function f on [a, b], Hă
older of order , is of bounded 1/-variation. Indeed,
this is a result of the bound
n

s1/α (f ; κ) ≤ K 1/α

i=1

|ti − ti−1 |α

1/α

= K 1/α (b − a),

(1.7)


valid for any partition κ = {ti }ni=0 of [a, b]. We have the following:
Proposition 1.1. If f and g : [a, b] R are Hă
older of orders and β respectively with α + β > 1, then the Riemann–Stieltjes integral (RS) ∫ab f dg
exists.
Proposition 1.1 is a special case of the following:
Proposition 1.2. If f and g : [a, b] → R are continuous, f ∈ Wp [a, b] and
g ∈ Wq [a, b] with 1 ≤ p < ∞, 1 ≤ q < ∞, and p−1 + q −1 > 1, then the
Riemann–Stieltjes integral (RS) ∫ab f dg exists.
Proposition 1.2, in turn, is a special case of Corollary 3.91 in light of
Definition 2.41 below.
If g is not of bounded 1-variation then ∫ab f dg is not a Lebesgue–Stieltjes
integral. If f is also not of bounded 1-variation, one cannot use integration by
parts to obtain a Lebesgue–Stieltjes integral. For any α with 0 < α < 1 there
exist functions g, Hă
older of order , which are not of bounded 1-variation.
Examples can be given by way of lacunary Fourier series, e.g. in the proof
of Theorem 3.75. So in defining integrals ∫ab f dg, neither the (refinement)
Riemann–Stieltjes nor the Lebesgue–Stieltjes integral is adequate in general.
If f and g are regulated functions, then there is an integral (originating in the


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1.1 How to Define ∫ab f dg?

5

paper [254] of W. H. Young) which works in the cases we have mentioned. A
function g on [a, b] is called regulated if for each t ∈ [a, b), the limit g(t+) :=
limu↓t g(u) exists, and for each s ∈ (a, b], the limit g(s−) := limu↑t g(u) exists.
If g has bounded p-variation for some 0 < p < ∞ then g is regulated (as

will be shown in Proposition 3.33). For any regulated function g on [a, b],
let ∆+ g(t) := g(t+) − g(t) if a ≤ t < b, and let ∆− g(s) := g(s) − g(s−)
if a < s ≤ b. A tagged partition ({ti }ni=0 , {si }ni=1 ) is called a Young tagged
partition if ti−1 < si < ti for each i = 1, . . . , n. Let f be any function on [a, b],
and let g be a regulated function on [a, b]. Given a Young tagged partition
τ = ({ti }ni=0 , {si }ni=1 ), the sum
SYS (f, dg; τ )
n

:=
i=1

f (ti−1 )∆+ g(ti−1 ) + f (si )[g(ti −) − g(ti−1 +)] + f (ti )∆− g(ti )

is called the Young–Stieltjes sum based on τ . The refinement Young–Stieltjes
integral of f with respect to g is defined and equals C ∈ R if for each ǫ > 0
there exists a partition λ of [a, b] such that |C − SYS (f, dg; τ )| < ǫ for each
Young tagged partition τ which is a refinement of λ. Then we let
b

(RYS)

f dg := lim SYS (f, dg; τ ) := C.
τ

a

If the refinement Riemann–Stieltjes integral exists then so does the refinement Young–Stieltjes integral with the same value, but not conversely (see
Proposition 2.18 below and the example following it). Let g be a nondecreasing function on [a, b], right-continuous on [a, b), and let µg be the Lebesgue–
Stieltjes measure on [a, b]. Then for any µg -measurable function f on [a, b],

(RYS) ∫ab f dg = (LS) ∫ab f dg whenever both integrals exist (see Propositions
2.27 and 2.28 below). The next fact follows from Corollary 3.91:
Proposition 1.3. If f ∈ Wp [a, b] and g ∈ Wq [a, b] with 1 ≤ p < ∞, 1 ≤
q < ∞, and p−1 + q −1 > 1, then the refinement Young–Stieltjes integral
(RYS) ∫ab f dg exists, and there is a constant Kp,q , depending only on p and
q, such that
b

(RYS)
a

f dg ≤ Kp,q f

[p]

g

(q) .

The integrals (A) ∫ab f dg have been defined for A = RS, RRS, and RYS
so far only when a < b. If a = b then we let ∫aa f dg := 0 for each of the three
integrals and call any function on a singleton regulated.
Interval functions
The Lebesgue–Stieltjes integral is essentially the Lebesgue integral with respect to a countably additive (signed) measure. Similarly, we can consider


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1 Introduction and Overview


extended Riemann–Stieltjes integrals with respect to a function µ defined on
sets, namely subintervals of a nonempty interval J, and thus to be called an
interval function on J. An interval function µ on J with values in a vector
space will be called additive if µ(A ∪ B) = µ(A) + µ(B) whenever A, B are
disjoint subintervals of J and their union A ∪ B also is an interval.
The following is a concrete variant of a general integral introduced by A.
N. Kolmogorov in [120]. Let J be a nonempty interval, open or closed at
either end. An ordered collection {Ai }ni=1 of disjoint nonempty subintervals
Ai of J is called an interval partition of J if their union is J and s < t for
all s ∈ Ai and t ∈ Aj whenever i < j. An interval partition which consists
only of open intervals and singletons is called a Young interval partition. If
{Ai }ni=1 is an interval partition of J then an ordered pair ({Ai }ni=1 , {si }ni=1 ) is
called a tagged interval partition of J and {si }ni=1 a set of tags for {Ai }ni=1 if
si ∈ Ai for i = 1, . . . , n. An interval partition A is a refinement of an interval
partition B, written A ⊐ B, if each interval in A is a subinterval of an interval
in B.
Similarly, a tagged interval partition T = (A, ξ) is a tagged refinement of
an interval partition B if A is a refinement of B. Let f be a function on J
and let µ be an additive interval function on J, both real-valued. For a tagged
interval partition T = ({Ai }ni=1 , {si }ni=1 ) of J, the sum
n

SK (f, dµ; T ) =

f (si )µ(Ai )
i=1

is called the Kolmogorov sum for f based on T . The Kolmogorov integral of
f with respect to µ is defined and equals C ∈ R if for each ǫ > 0 there exists

an interval partition A of J such that |C − SK (f, µ; T )| < ǫ for each tagged
interval partition T of J which is a tagged refinement of A. Then we let
= f dµ := lim SK (f, dµ; T ) := C.
J

T

The Kolmogorov integral can be related to the refinement Young–Stieltjes
integral as follows. An interval function µ on [a, b] will be called upper continuous if µ(An ) → µ(A) for any sequence of subintervals A1 , A2 , . . . of
[a, b] such that An ↓ A. For any regulated function g on [a, b], there exists an additive upper continuous interval function µg on [a, b] such that
µg ((s, t)) = g(t−) − g(s+) and µg ({t}) = g(t+) − g(t−) for s < t in [a, b], setting g(a−) := g(a) and g(b+) := g(b) in this case. For a = b let µg ({a}) := 0.
Let g be a regulated function on [a, b], and let µg be the corresponding additive upper continuous interval function on [a, b]. There is a 1–1 correspondence
between tagged Young partitions τ of [a, b] and tagged Young interval partitions T of [a, b], with SYS (f, dg; τ ) ≡ SK (f, dµg ; T ). The existence of the
Kolmogorov integral with respect to an upper continuous additive interval
function depends only on Kolmogorov sums which are based on tagged Young
interval partitions, as shown in Proposition 2.25 below. Therefore


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1.2 Some Integral and Differential Equations

7

b

(RYS)

f dg = = f dµg
a


[a,b]

if either side is defined. Note that in an integral (RYS) ∫ab f dg, if for some
t ∈ (a, b), g(t) has a value different from g(t−) or g(t+), the value g(t) has no
influence on the value of the integral.
In this book we mainly use interval functions and the Kolmogorov integral.
The inequality of Proposition 1.3 extends to the Kolmogorov integral using
the p-variation for interval functions defined as follows. Let µ be an additive
interval function on a nonempty interval J and let 0 < p < ∞. For an interval
n
partition A = {Ai }ni=1 of J, let sp (µ; A) := i=1 |µ(Ai )|p . The p-variation of
µ on J is an interval function vp (µ) = vp (µ; ·) on J defined by
vp (µ; A) := sup sp (µ; A)
A

if A is a nonempty subinterval of J, where the supremum is over all interval
partitions A of A, or as 0 if A = ∅. We say that µ has bounded p-variation if
vp (µ; J) < ∞. For a subinterval A ⊂ J, let Vp (µ; A) := vp (µ; A)1/p . The class
of all additive and upper continuous interval functions on J with bounded
p-variation is denoted by AI p (J). The following analogue of Proposition 1.3
for the Kolmogorov integral is a special case of Corollary 3.95.
Theorem 1.4. If µ ∈ AI p [a, b] and f ∈ Wq [a, b] with 1 ≤ p < ∞, 1 ≤ q < ∞,
and p−1 + q −1 > 1, then the Kolmogorov integral =[a,b] f dµ exists, and there
is a constant Kp,q depending only on p and q such that
= f dµ] ≤ Kp,q f

[a,b],[q] Vp (µ; [a, b]).

[a,b]


In Chapter 2, integrals will be defined where integrands f (and for bilinear
integrals also g) and interval functions µ can all have values in Banach spaces.

1.2 Some Integral and Differential Equations
Consider a linear integral equation
t

f dh,

f (t) = 1 + (RYS)
0

0 ≤ t ≤ 2,

with respect to a function h : [0, 2] → R, and/or a linear Kolmogorov integral
equation
f (t) = 1 + = f dµ,
0 ≤ t ≤ 2,
(1.8)
[0,t]

where µ(A) := δ1 (A) := 1A (1) for any interval A ⊂ [0, 2] and h(t) := 1[1,∞) (t).
Either equation gives f (1) = 1 + f (1), a contradiction. If instead we take the
integral equation


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1 Introduction and Overview


f (t) = 1 + = f dµ,
[0,t)

0 ≤ t ≤ 2,

then it has the solution f (t) = 1 for 0 ≤ t ≤ 1, f (t) = 2 for 1 < t ≤ 2.
Equations of the given form can be solved rather generally; see Section 9.11.
Product integration
Consider a linear kth order ordinary differential equation
dk x(t)
dk−1 x(t)
+
u
(t)
+ · · · + u0 (t)x(t) = v(t),
k−1
dtk
dtk−1

a ≤ t ≤ b,

(1.9)

where for the present, x and the coefficients uj and v are real-valued. As is
often done in differential equations, one can write an equivalent first-order
linear vector and matrix differential equation
df (t)/dt = C(t) · f (t),

(1.10)


where f (·) is the (k + 1) × 1 column vector and C(·) is the (k + 1) × (k + 1)
matrix-valued function defined respectively by




0
0
0
0
...
0
1
 0
0
1
0
...
0 

 x(t) 




0
0
1
...

0 
′

 and C(t) =  0
x
(t)
f (t) = 
 ... ...... ...... ...... ...... ...... 




 ... 
 0
0
0
0
...
1 
x(k−1) (t)
v −u0 −u1 −u2 . . . −uk−1
with x(j) (t) := dj x(t)/dtj , v := v(t), and uj := uj (t) for j = 0, 1, . . . , k − 1.
It is easy to check that for two 3 × 3 matrices A, B of the form of C, so that
k = 2, we have AB = BA if and only if A = B. The same is true for any k ≥ 2:
consider the next-to-last row of the products. Thus commuting matrices will
be obtained only for differential equations with constant coefficients if k ≥ 2.
If (1.10) holds at a point t, then
f (t + s) − f (t) = sC(t) · f (t) + o(s)
f (t + s) = I + sC(t) · f (t) + o(s)


as s ↓ 0,
as s ↓ 0,

or
(1.11)

where I is the (k +1)×(k +1) identity matrix. Suppose that C(·) is continuous
on an interval [a, b] and let h(t) := ∫at C(s) ds, so that h is a C 1 function with
h(a) = 0. Let a = t0 < t1 < · · · < tn = t be a partition of [a, t] where
a < t ≤ b. Then (1.11) implies that approximately
.
f (t) = (I + h(tn ) − h(tn−1 )) · · · (I + h(t1 ) − h(t0 ))f (a)

(1.12)

for a fine enough partition. Taking a limit of such products (without the f (a)
t
factor) as the mesh of the partition goes to 0, we get a matrix called a (I+dh),


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1.3 Basic Assumptions

9

the product integral of h from a to t. The product integral will give us the
value of the solution of the differential equation (1.9) in terms of the values
of x(·) and its derivatives through order k − 1 at t = a, via
t


f (t) =

(I + dh) f (a).
a

If the increments of h commute then the product integral with respect to
h over an interval [a, t] is the exponential exp{h(t) − h(a)}, as follows from
Theorem 9.40 below and continuity of h. We will show in Section 9.12 that
f may have values in a Banach space X, and so h, as well as the product
integral with respect to h, will have values in the Banach algebra L(X, X) of
bounded operators from X into itself.
So far, h has been a C 1 point function. In Chapter 9, we will see how
the product integral can be defined for interval functions with values in any
Banach algebra, of bounded p-variation for 1 ≤ p < 2. The product integral
will give, in Sections 9.11 and 9.12, solutions of integral equations, where the
integrals in the equations are Kolmogorov integrals, defined briefly in Section
1.1 and treated more fully in Section 2.3.

1.3 Basic Assumptions
Let K be either the field R of real numbers or the field C of complex numbers.
Let X, Y and Z be Banach spaces over K. The norm on each will be denoted
by · . For intuition, have in mind the case X = Y = Z = K with x = |x|
for all x. Let B(·, ·) be a bounded bilinear operator from X × Y into Z, where
“bounded” means that for some M < ∞,
B(x, y) ≤ M x

y

(1.13)


for all x ∈ X and y ∈ Y . In the case X = Y = Z = K we will take
B(x, y) ≡ xy. If (1.13) holds for a given M we will say that B is M -bounded.
If the three spaces X, Y, and Z are all different then by changing the norm
to an equivalent one on any one of the spaces by a constant multiple, we can
assume M = 1. For a fixed B we will write x·y := B(x, y). For later reference
we summarize some assumptions:
X, Y, Z are Banach spaces over K,
X×Y ∋ (x, y) → x·y ∈ Z is 1-bounded and bilinear.

(1.14)

For example, let B be a Banach algebra over K with a norm · , as treated
in Chapter 4. Then we can take X = Y = Z = B and · as the multiplication
in B. By an equivalent renorming of B one can take the multiplication to be
1-bounded, as will be seen in Theorem 4.8.


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10

1 Introduction and Overview

Reversing the order of integrand and integrator
Under the assumption (1.14), we will be giving various definitions of integrals
b
a

b

f ·dg ≡


B(f, dg) and
a

= f ·dµ ≡ = B(f, dµ),
J

J

where −∞ < a ≤ b < +∞, f : [a, b] → X, g : [a, b] → Y , J is a subinterval of
[a, b] and µ is an interval function on [a, b] with values in Y . Then, for each
definition of integral ∫ab f ·dg, we will have the corresponding definition
b

a

b

b

df ·g ≡

B(df, g) :=

˜ df ),
B(g,

(1.15)

a


a

˜ x) := B(x, y) is a bounded bilinear operator: Y ×X → Z and the
where B(y,
integrals on the left are defined if and only if the integral on the right is.
The integrals ∫ab f ·dg with a < b will be defined as limits of certain sums, but
the sums for f and g will in general not be symmetric in f and g, even if
X = Y and B(y, x) ≡ B(x, y). Likewise, given a definition of =[a,b] f ·dµ and
an X-valued interval function ν on [a, b], we will write
˜ dν).
= dν·g ≡ = B(dν, g) := = B(g,
J

J

(1.16)

J

1.4 Notation and Elementary Notions
Spaces of operators
Let X and Y be normed linear spaces. A linear function T from X into Y is
called a bounded linear operator iff
T

:= sup{ T x : x ∈ X, x ≤ 1} < ∞.

(1.17)


Then T is called the operator norm of T . The set of all bounded linear
operators from X into Y will be called L(X, Y ). It is easily seen to be a
normed linear space with the operator norm. If Y is a Banach space, then so
is L(X, Y ).
Spaces of bounded functions
For a function f from a nonempty set S into a normed space X, let
f

sup

:=

f

S,sup

:= sup{ f (x) : x ∈ S}.

Restricted to functions for which it is finite, i.e., bounded functions, · sup
is called the sup norm. The normed space of all bounded X-valued functions
on X is denoted by ℓ∞ (S; X). Also, the oscillation of f on S is defined by
OscS (f ) := Osc (f ; S) := sup

f (s) − f (t) : s, t ∈ S .


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1.4 Notation and Elementary Notions

11


Hă
older classes
Let X, Y be normed spaces, let U be a subset of X with more than one
element, and let 0 < α ≤ 1. A function f : U → Y is said to be Hă
older of
order , or simply -Hă
older, if
f

(H )

:= f
:= sup

U,(H )

f (x) − f (y) / x − y

α

: x, y U, x = y < . (1.18)

An -Hăolder function on U is clearly continuous on U and will sometimes
be called -Hă
older continuous. The class of all -Hăolder functions from U
into Y is denoted by Hα (U ; Y ). In the case U = X = Y = R we write
Hα = Hα (R; R).
Banach spaces of functions
Let S be a nonempty set, and let X be a Banach space over K. A set F of

X-valued functions on S is a vector space over K if it is a vector space with
respect to pointwise operations on S, that is, for f, g ∈ F, a scalar r ∈ K, and
any s ∈ S,
(rf )(s) = rf (s)

and

(f + g)(s) = f (s) + g(s).

For example, the set X S of all X-valued functions on S is a vector space. If
F ⊂ X S and · is a norm on F, then (F, · ) will be called a Banach space
of X-valued functions on S iff F is a vector space and (F, · ) is a Banach
space. If also X = K then (F, · ) will be called a Banach space of functions.
Intervals
An interval in R is a set of any of the following four forms: for −∞ ≤ u ≤ v ≤
+∞,
(u, v) := {t ∈ R : u < t < v},
[u, v) := {t ∈ R : u ≤ t < v}, with −∞ < u,
(u, v] := {t ∈ R : u < t ≤ v}, with v < +∞, and
[u, v] := {t ∈ R : u ≤ t ≤ v}, with −∞ < u ≤ v < +∞.
For each of the four cases, if the interval is nonempty, u is called its left
endpoint and v its right endpoint. For any u ∈ R, [u, u] = {u} is a singleton
and (u, u] = [u, u) = (u, u) = ∅. An interval is called bounded if it is empty or
its left and right endpoints are finite. An interval will be called nondegenerate
if it contains more than one point. Let J be an interval in R. The class of
all subintervals of J will be denoted by I(J). The subclass of I(J) consisting
of nonempty open intervals and singletons will be denoted by Ios (J). If J =
[a, b] then we write I[a, b] and Ios [a, b]. For example, if a < b, Ios [a, b] =
{(u, v), {u}, {v} : a ≤ u < v ≤ b}.



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12

1 Introduction and Overview

An interval A will be called right-open at v or in symbols A = [[·, v) if
A = [u, v) or (u, v) for some u < v. Similarly, A will be called right-closed at
v, or A = [[·, v], if J = [u, v] for some u ≤ v or A = (u, v] for some u < v. An
interval A will be called left-open at u or A = (u, ·]] if A = (u, v) or (u, v] for
some u < v, and A will be called left-closed at u, or A = [u, ·]], if A = [u, v]
for some u ≤ v or A = [u, v) for some u < v. For a ≤ u < v ≤ b, any of
the four intervals [u, v], (u, v], [u, v) or (u, v) will be denoted by [[u, v]]. Here
[[−∞, v]] or [[u, +∞]] will always mean (−∞, v]] or [[u, +∞), respectively. For
two disjoint nonempty intervals A and B, A ≺ B will mean that s < t for all
s ∈ A and t ∈ B.
Point partitions
Let (S, <) be any linearly ordered set containing more than one point. Then
a point partition κ = {ti }ni=0 of S is a finite sequence of elements of S such
that (i) t0 < t1 < · · · < tn , (ii) if S has a smallest element a, then t0 = a,
and (iii) if S has a largest element b, then tn = b. Let PP (S) denote the set
of all point partitions of S.
Thus for a nondegenerate interval J ⊂ R, κ = {ti }ni=0 ⊂ J is a point
partition of J if t0 < t1 < · · · < tn , and if J = [a, b], a closed bounded interval,
then t0 = a and tn = b. At the beginning of Section 1.1 point partitions of
the closed interval J = [a, b] were defined and called partitions. Most of the
further terminology related to point partitions was already given near the
beginning of Section 1.1 and is repeated here for the reader’s convenience.
For κ := {ti }ni=0 ∈ PP [a, b] with −∞ < a < b < +∞, the mesh of κ is
|κ| := max1≤i≤n (ti − ti−1 ). A point partition κ is a refinement of a point

partition λ if λ ⊂ κ as a set. Let κ = {ti }ni=0 be a partition of [a, b], and
let si ∈ [ti−1 , ti ] for i = 1, . . . , n. Then τ = ({ti }ni=0 , {si }ni=1 ) is called a
tagged partition of [a, b], and τ is a tagged refinement of a point partition
λ if κ is a refinement of λ. We will also say that the tagged partition τ
consists of the tagged intervals ([ti−1 , ti ], si ), i = 1, . . . , n. If a tagged partition
τ = ({ti }ni=0 , {si }ni=1 ) is such that si ∈ (ti−1 , ti ) for i = 1, . . . , n, then τ is
called a Young tagged point partition. The mesh |τ | is defined as |κ|.
Interval partitions
Let J be a nonempty interval in R. Recall the terminology defined in the subsection “Interval functions” of Section 1.1. Let IP (J) be the set of all interval
partitions of J. If J is a bounded, nondegenerate interval [t0 , tk ] and A =
{Ai }ni=1 is a Young interval partition {{t0 }, (t0 , t1 ), {t1 }, . . . , (tk−1 , tk ), {tk }}
of J, then {(ti−1 , ti )}ki=1 sometimes will be written instead of A and we denote
by ({(ti−1 , ti )}ki=1 , {ui }ki=1 ) a corresponding tagged Young interval partition.
(The singletons {ti } with their uniquely determined tags ti are omitted from
the notation. Here n = 2k + 1.) Similarly, if J is left-open and/or right-open
the same notation will be used where now {t0 } ∈ A and/or {tk } ∈ A.