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## Über dieses Buch

to the English Translation This is a concise guide to basic sections of modern functional analysis. Included are such topics as the principles of Banach and Hilbert spaces, the theory of multinormed and uniform spaces, the Riesz-Dunford holomorphic functional calculus, the Fredholm index theory, convex analysis and duality theory for locally convex spaces. With standard provisos the presentation is self-contained, exposing about a h- dred famous "named" theorems furnished with complete proofs and culminating in the Gelfand-Nalmark-Segal construction for C*-algebras. The first Russian edition was printed by the Siberian Division of "Nauka" P- lishers in 1983. Since then the monograph has served as the standard textbook on functional analysis at the University of Novosibirsk. This volume is translated from the second Russian edition printed by the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences· in 1995. It incorporates new sections on Radon measures, the Schwartz spaces of distributions, and a supplementary list of theoretical exercises and problems. This edition was typeset using AMS-'lEX, the American Mathematical Society's 'lEX system. To clear my conscience completely, I also confess that := stands for the definor, the assignment operator, signifies the end of the proof.

## Inhaltsverzeichnis

### Chapter 1. An Excursion into Set Theory

Abstract
Let A and B be sets and let F be a subset of the product A × B := {(a, b) : aA, bB}. Then F is a correspondence with the set of departure A and the set of arrival B or just a correspondence from A (in)to B.

### Chapter 2. Vector Spaces

Abstract
In algebra, in particular, modules over rings are studied. A module X over a ring A is defined by an abelian group (X, +) and a representation of the ring A in the endomorphism ring of X which is considered as left multiplication : A × XX by elements of A. Moreover, a natural agreement is presumed between addition and multiplication. With this in mind, the following phrase is interpreted: “A module X over a ring A is described by the quadruple (X, A, +, ·).” Note also that A is referred to as the ground ring of X.

### Chapter 3. Convex Analysis

Abstract
Let Г be a subset of 𝔽2. A subset U of a vector space is a Г-set in this space (in symbols, U ∈ (Г)) if $$\left( {{\lambda _1},{\lambda _2}} \right) \in \Gamma \Rightarrow {\lambda _1}U + {\lambda _2} \subset U$$ .

### Chapter 4. An Excursion into Metric Spaces

Abstract
A mapping d: X 2 → ℝ+ is a metric on X if
(1)
$$d\left( {x,y} \right) = 0 \Leftrightarrow x = y$$;

2)
$$d\left( {x,y} \right) = d\left( {y,x} \right)\;\left( {x,y \in X} \right)$$;

(3)
$$d\left( {x,y} \right) \le d\left( {x,z} \right) + d\left( {z,y} \right)\;\left( {x,y,z \in X} \right)$$.

### Chapter 5. Multinormed and Banach Spaces

Abstract
Let X be a vector space over a basic field 𝔽 and let p: X → ℝ be a seminorm. Then
(1)
dom p is a subspace of X;

(2)
p(x) ≥ 0 for all x ∈ X;

(3)
the kernel ker p≔ {p = 0} is a subspace in X;

(4)
the sets $${{\mathop{B}\limits^{ \circ } }_{p}}: = \left\{ {p < 1} \right\}$$ and B p ≔ {p ≤ 1} are absolutely convex; moreover, p is the Minkowski functional of every set B such that $${{\mathop{B}\limits^{ \circ } }_{p}} \subset B \subset {{B}_{p}};$$

(5)
X = dom p if and only if $${{\mathop{B}\limits^{ \circ } }_{p}}$$ is absorbing.

### Chapter 6. Hilbert Spaces

Abstract
Let H be a vector space over a basic field 𝔽. A mapping f: H 2 → 𝔽 is a hermitian from on H provided that
(1)
the mapping f(·, y) : x ↦ f(x, y) belongs to H# for every y in Y;

(2)
f(x, y) = f(y, x)* for all x, y ∈ H, where λ ↦ λ* is the natural involution in 𝔽; that is, the taking of the complex conjugate of a complex number.

### Chapter 7. Principles of Banach Spaces

Abstract
Let U be aconvex set with nonempty interior in a multinormed space: int U ≠ ∅. Then
(1)
0 ≤ α < 1 ⇒ αcl U + (1 - α) int U ⊂ int U;

(2)
U = int U;

(3)
cl U = cl int U;

(4)
int cl U = int U.

### Chapter 8. Operators in Banach Spaces

Abstract
Let X be a Banach space. A subset Λ of the ball B x′ in the dual space X′ is called norming (for X) if ‖x‖ = sup{|l (x)| : l ∈ Λ} for all xX. If each subset U of X satisfies sup ‖U‖ < +∞ on condition that sup{|l(u)|: uU} < +∞ for all l ∈ Λ, then Λ is a fully norming set.

### Chapter 9. An Excursion into General Topology

Abstract
Let X be a set. A mapping τ : X𝒫(𝒫(X)) is a pretopology on X if
(1)
x ∈ X ⇒ t(x) is a filter on X;

(2)
x ∈ X ⇒ t(x) ⊂ fil {x}

### Chapter 10. Duality and Its Applications

Abstract
Let (X, 𝔽, +, ·) be a vector space over a basic field 𝔽. A topology τ on X is a topology compatible with vector structure or, briefly, a vector topology, if the following mappings are continuous:
$$\begin{gathered} + :\left( {X \times X,\tau \times \tau } \right) \to \left( {X,\tau } \right), \hfill \\ \cdot :\left( {\mathbb{F} \times X,{\tau _\mathbb{F}} \times \tau } \right) \to \left( {X,\tau } \right). \hfill \\ \end{gathered}$$