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1992 | Book

Set Algebra and Function Lattices Read chapter Read first chapter

Problems in Real and Complex Analysis

Author: Bernard R. Gelbaum

Publisher: Springer New York

Book Series : Problem Books in Mathematics

Included in: Professional Book Archive

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About this book

In the pages that follow there are: A. A revised and enlarged version of Problems in analysis (PIA) . (All typographical, stylistic, and mathematical errors in PIA and known to the writer have been corrected.) B. A new section COMPLEX ANALYSIS containing problems distributed among many of the principal topics in the theory of functions of a complex variable. C. A total of 878 problems and their solutions. D. An enlarged Index/Glossary and an enlarged Symbol List. Notational and terminological conventions are to be found for the most part under Conventions at the beginnings of the chapters. Spe­ cial items not included in Conventions are completely explained in the Index/Glossary. The audience to which the current book is addressed differs little from the audience for PIA. The background of the reader is assumed to include a knowledge of the basic principles and theorems in real and complex analysis as those subjects are currently viewed. The aim of the problems is to sharpen and deepen the understanding of the mechanisms that underlie modern analysis. I thank Springer-Verlag for its interest in and support of this project. State University of New York at Buffalo B. R. G. v Contents The symbol alb under Pages below indicates that the Problems for the section begin on page a and the corresponding Solutions begin on page b. Thus 3/139 on the line for Set Algebra indicates that the Problems in Set Algebra begin on page 3 and the corresponding Solutions begin on page 139.

Table of Contents

Frontmatter

Real Analysis: Problems

Frontmatter
1. Set Algebra and Function Lattices
Abstract
When n is a cardinal number, the phrase “n objects” signifies n pairwise different objects.
Bernard R. Gelbaum
2. Topology, Limits, and Continuity
Abstract
A topological space is a pair (X, T) (or simply X) consisting of a set X and a subset T of 2 X : T is the topology for X. The elements of T are the open sets of X, ϕ, X ∊ T, and T is closed with respect to the formation of arbitrary unions and finite intersections.
Bernard R. Gelbaum
3. Real- and Complex-valued Functions
Abstract
A map ϕ: of a convex open subset K of a vector space V is convex iff for all t in [0,1] and any x and y in K
$$ \varphi (t{\text{x + (1 - }}t{\text{)y}}) \leqslant t\varphi ({\text{x}}) + (1 - t)\varphi ({\text{y}}) $$
. When K is an open interval in ℝ and a ∈ K, a line l through (a, φ(a)) is a supporting line iff l lies below the graph of φ: {(p, q) ∈ l} ⇒ {q ≤ φ(p)}.
Bernard R. Gelbaum
4. Measure and Topology
Abstract
For (X, Sβ,μ), a Borel set E is inner resp. outer regular iff resp. E is regular iff E is both inner and outer regular; μ is inner (outer) regular iff every Borel set is inner (outer) regular. (Similar conventions for regular apply to arbitrary nonnegative set functions.)
Bernard R. Gelbaum
5. Measure Theory
Abstract
When (X, S, μ) is signed μ is the difference of two positive measures such that for every E in S, at least one of μ±(E) is in. There are in S two sets P ± such that for E in S, μ± (E) = μ (P ± ∩ E) and \(\left| \mu \right|\mathop = \limits^{{\text{def}}} \mu ^ + + \mu ^ - \).
Bernard R. Gelbaum
6. Topological Vector Spaces
Abstract
The topological vector space V (over a topological field K) is locally convex iff each element of some neighborhood base at O, is convex. A subset B of V is bounded iff for every neighborhood U of O and some positive t. When some open subset of V is bounded, V is locally bounded. A subset S of V is circled iff and for every complex number z such that ||z| ≤ 1 and every x in S. For a circled bounded neighborhood of U of O, the associated Minkowski functional M U is:
$$ M_U :V \mathrel\backepsilon {\text{x}} \mapsto \inf \{ \alpha :\alpha \geqslant 0,{\text{x}} \in \alpha U\} $$
. A map p : V ∋ x ↦ p(x) ∈ [0,∞) is a quasinorm iff for some k in [1,∞): a) p(x + y) ≤ k (p(x) + p(y)); b) p|t|x) When k = 1, p is a seminorm. A seminorm p is a norm iff {p(x) = 0} ⇔ {x = O}. A subset R of V is radial or absorbing iff for every finite subset F of V and some real number r(F), λR:⊃ F if |λ| ≥ r(F).
Bernard R. Gelbaum

Complex Analysis: Problems

Frontmatter
7. Elementary Theory
Abstract
For each subset S of, the set of complex conjugates of elements of S is denoted. On the other hand, the topological closure of S, i.e., the intersection of all closed sets containing S, is denoted S c .
Bernard R. Gelbaum
8. Functions Holomorphic in a Disc
Abstract
For f in H(U) and t in [0, 2π),
$$ f*(t)\mathop = \limits^{def} \left\{ {_0^{\lim _{r \uparrow 1} f\left( {re^{it} } \right)} } \right.\,{\text{when}}\,{\text{the limit}}\,{\text{exists}}\,{\text{otherwise}} $$
is the radial limit function. The (Hardy class)
$$ \left\{ {f:f \in H(U),_{z \in U}^{\sup \left| {f(z)} \right| < \infty } } \right\} $$
isH ∞.
Bernard R. Gelbaum
9. Functions Holomorphic in a Region
Abstract
The sets \(\left\{ {z:\Im (z)_ < ^ > 0} \right\}\) are denoted Π±.
Bernard R. Gelbaum
10. Entire Functions
Abstract
Unless the contrary is indicated, every function introduced in this Chapter is entire. An entire function that is not a polynomial is transcendental.
Bernard R. Gelbaum
11. Analytic Continuation
Abstract
When Ω1,Ω2 are two regions such that and, i = 1, 2, while f1(z) = f2(z) in Ω3 then the function element (f2, Ω2) resp. (f1, Ω1) is an immediate analytic continuation of the function element (f1, Ω1) resp. (f 2,Ω2). When Ω i , 1 ≤ i ≤n, is a finite sequence of regions such that
$$ \Omega _i \cap \Omega _{i + 1} \ne \not 0,1 \leqslant i \leqslant n - 1,\,f_i \in H(\Omega _i ),1 \leqslant i \leqslant n $$
, and each (f i+1, Ω i+1) is an immediate analytic continuation of (f i , Ω i ) then (f n , Ω n ) is an analytic continuation of (f1, Ω1).
Bernard R. Gelbaum
12. Singularities
Abstract
Each function introduced in this Chapter is assumed to be differentiable at all points except those specified as singularities. A singularity need not be isolated, e.g., the singularity of \( \frac{1}{{\sin \left( {\frac{1}{z}} \right)}} \) at zero.
Bernard R. Gelbaum
13. Harmonic Functions
Abstract
When Ω is an open subset of, h(x + iy) in C 2(Ω), is harmonic iff \(\Delta h\mathop = \limits^{def} h_{xx} + h_{yy} = 0\) in Ω. The operator ▵ is the Laplacian and the set of all functions harmonic in Ω is denoted L(Ω). When Ω is a region and h is (f) for some \(f\mathop = \limits^{def} u + iv\) in H(Ω) then v is an harmonic conjugate of h.
Bernard R. Gelbaum
14. Families of Functions
Abstract
A set F of functions in H(Ω) is normal iff every sequence in contains a subsequence converging uniformly on compact sets of Ω to a function [not necessarily in H(Ω)]. When the ranges of the elements F of are regarded in and convergence is regarded as taking place in, where ∞ is a legitimate limit, there is a corresponding notion of spherical normality.
Bernard R. Gelbaum
15. Convexity Theorems
Abstract
In such disparate contexts as the Hahn-Banach theorem, seminorms, Kol-mogorov’s theorem (cf. 6.162), von Neumann’s theory of almost periodic functions on groups (cf. 2.93), his theory of games [NM] and linear programming, Jensen’s inequality, Hadamard’s three-lines/three-circles theorems, the Hausdorff-Young theorem, probability theory, etc., the rôle played by convexity is central. The following discussion attempts to illustrate that role in complex analysis.
Bernard R. Gelbaum

Real Analysis: Solutions

Frontmatter
1. Set Algebra and Function Lattices
Abstract
References to items in SOLUTIONS are tagged with the prefix s to distinguish them from items in PROBLEMS.
Bernard R. Gelbaum
2. Topology, Limits, and Continuity
Abstract
a) If A contains the range of the net n converging to a then for each for each N (a), n is eventually in N(a) whence N(a) ∩A ≠ ø and so a ∈ Ā. If Λ is the diset N(a) then the range of the net n : Λ ∋ N ↦ n(N) ∈ N ∩ A is contained in A and n converges to a
Bernard R. Gelbaum
3. Real- and Complex-valued Functions
Abstract
For each n in D and some index m greater than n, s m > s n-1. Let n′ be the least such m. If n < p < n′ then
$$ s_{n'} - s_{n - 1} = s_{n'} - s_{p - 1} + s_{p - 1} - s_{n - 1} > 0 $$
whereas S p-1-S n-1≤ 0, and so S n′ - S p-1>. In other words, each p in (n, n′) is distinguished, i.e., the numbers n, n + 1,…, n′ - 1 constitute a block or a part of a block.
Bernard R. Gelbaum
4. Measure and Topology
Abstract
a) Because L(f) ∈ (u, v), if y = m(x - a) + φ(a) is the equation of a supporting line through (a, φ(a)), for xin X,
$$ \begin{gathered} m(f(x) - a) + \varphi (a) \leqslant \varphi of(x) \hfill \\ \end{gathered} $$
.
Bernard R. Gelbaum
5. Measure Theory
Abstract
Let μ be a finite measure. If μ is positive and
$$ M\mathop = \limits^{{\text{def}}} \sup \left\{ {\mu \left( E \right)} \right.:Emeasurable\left. \right\} $$
then for some sequence {E n}n∈N of measurable sets, μ(E n)↑ M. It may be assumed that {E n}n∈N is a monotonely increasing sequence, whence lim \(n \to \infty En\mathop = \limits^{{\text{def}}} E\) is measurable and M = lim n→∞μ (E n) =μ(E) < ∞.
Bernard R. Gelbaum
6. Topological Vector Spaces
Abstract
Because E≠(f, 0) is σ-finite, E= lim n→∞ for some sets E n of finite measure. Hence
$$ \int {_X f(x)d\mu (x) = \int {_E f(x)} d\mu (x) = _{n \to \infty }^{\lim } \int {_{E_n } } f(x)d\mu (x) \leqslant a} $$
.
Bernard R. Gelbaum

Complex Analysis: Solutions

Frontmatter
7. Elementary Theory
Abstract
A circle on S is the intersection of ∑ with a plane Π for which the equation is aξ + bη) + cζ, = a 2 + b 2 + c 2. The plane Π and ∑ intersect iff a 2 + b 2 + c 2 ≤ 1. The equation ξ2 + η2 + ζ2 = 1 and the formulae for the coordinates of \(\Theta \left( {\xi ,\eta ,\zeta \mathop = \limits^{{\text{def}}} (x,y)} \right)\) lead to the equation
$$ \left( {a^2 + b^2 + c^2 - c} \right)\left( {x^2 + y^2 } \right) - 2ax - 2by + a^2 + b^2 + c^2 + c = 0 $$
representing a circle in ℂ or, if a 2 + b 2 + c 2 = c, a straight line in ℂ. The latter circumstances imply that Π passes through (0,0,1). The reasoning is reversible and leads from a circle in Π to a circle on Σ\ {(0, 0, 1)} or from a straight line in ℂ to a circle passing through (0, 0, 1) on Σ.
Bernard R. Gelbaum
8. Functions Holomorphic in a Disc
Abstract
If f ∈ H(U) then f′(0) = 0 and f′(0) = 1 according as f′(0) is calculated by using the zeros of f or the values of 1/n where f (1/n) ≠ 0.
Bernard R. Gelbaum
9. Functions Holomorphic in a Region
Abstract
a) The function \(g(z)\mathop = \limits^{{\text{def}}} f(z) - f(z) - f'(a)z\) is such that g′(a) = 0 whence g is not injective near a, i.e., near a there are two points b, c such that g(b) = g(c), whence the conclusion.
Bernard R. Gelbaum
10. Entire Functions
Abstract
Because
$$ {z} \in \varepsilon $$
and |g(z)| is bounded, Liouville’s theorem implies g is a constant. Since lim |z|→∞g(z)≡f(0).
Bernard R. Gelbaum
11. Analytic Continuation
Bernard R. Gelbaum
12. Singularities
Bernard R. Gelbaum
13. Harmonic Functions
Abstract
If \(g\mathop = \limits^{{\text{def}}} u + iv\), the chain rule for calculating partial derivatives applies to h(u, v). Note that sinee g ∈ H(Ω), the Cauchy-Riemann equations imply that both u and v are harmonic in Ω.
Bernard R. Gelbaum
14. Families of Functions
Abstract
Some subsequence {f nk }k∈N converges uniformly on compact subsets of Ω. Thus each of the sequences \(S_p \mathop = \limits^{{\text{def}}} \left\{ {f_{n_k }^{2^p } } \right\}_{k \in {\text{N}}} {\text{,}}\,p \in \mathbb{Z}^ + \) converges on compact subsets of Ω. Furthermore, S p+1 ⊂ S p . The diagonal sequence \(S\mathop = \limits^{{\text{def}}} \left\{ {f_{n_p }^{2^p } } \right\}p \in {\text{N}}\) is a subset of S 0.
Bernard R. Gelbaum
15. Convexity Theorems
Abstract
a) If f is monotonely increasing resp. decreasing then
$$ _{x \in [a,b]}^{\sup } \,f(x) = f(b)\,{\text{resp}}{\text{.}}\,_{x \in [a,b]}^{\sup } \,f(x) = f(a) $$
If f is convex then the result 3.8 implies the conclusion. b) For exampIe, In x satisfies the hypothesis hut In x is strictly concave.
Bernard R. Gelbaum
Backmatter
Metadata
Title
Problems in Real and Complex Analysis
Author
Bernard R. Gelbaum
Copyright Year
1992
Publisher
Springer New York
Electronic ISBN
978-1-4612-0925-6
Print ISBN
978-1-4612-6949-6
DOI
https://doi.org/10.1007/978-1-4612-0925-6