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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.

Inhaltsverzeichnis

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, XT, 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 aK, 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

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 Ω12 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 22). When Ω i , 1 ≤ in, 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

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 ∋ Nn(N) ∈ NA 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

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 fH(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

Without Abstract
Bernard R. Gelbaum

12. Singularities

Without Abstract
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 gH(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

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