Functional analysis : entering Hilbert space / Vagn Lundsgaard Hansen.

Format:

Book

Author/Creator:

Hansen, Vagn Lundsgaard.

Language:

English

Subjects (All):

Functional analysis.

Hilbert space.

Physical Description:

x, 136 pages ; 24 cm

Place of Publication:

New Jersey : World Scientific, [2006]

Summary:

This book presents basic elements of the theory of Hilbert spaces and operators on Hilbert spaces, culminating in a proof of the spectral theorem for compact, self-adjoint operators on separable Hilbert spaces. It exhibits a construction of the space of p[superscript th] power Lebesgue integrable functions by a completion procedure with respect to a suitable norm in a space of continuous functions, including proofs of the basic inequalities of Holder and Minkowski. The L[superscript p]-spaces thereby emerges in direct analogy with a construction of the real numbers from the rational numbers. This allows grasping the main ideas more rapidly. Other important Banach spaces arising from function spaces and sequence spaces are also treated. Book jacket.

Contents:

Preliminary Notions 1

1 Basic Elements of Metric Topology 5

1.1 Metric spaces 5

1.2 The topology of a metric space 9

1.3 Completeness of metric spaces 11

1.4 Normed vector spaces 15

1.5 Bounded linear operators 18

2 New Types of Function Spaces 23

2.1 Completion of metric spaces and normed vector spaces 23

2.2 The Weierstrass Approximation Theorem 28

2.3 Important inequalities for p-norms in spaces of continuous functions 32

2.4 Construction of L[superscript p]-spaces 36

2.4.1 The L[superscript p]-spaces and some basic inequalities 36

2.4.2 Lebesgue measurable subsets in R 39

2.4.3 Smooth functions with compact support 42

2.4.4 Riemann integrable functions 43

2.5 The sequence spaces l[superscript p] 45

3 Theory of Hilbert Spaces 49

3.1 Inner product spaces 49

3.2 Hilbert spaces 54

3.3 Basis in a normed vector space and separability 55

3.3.1 Infinite series in normed vector spaces 55

3.3.2 Separability of a normed vector space 56

3.4 Basis in a separable Hilbert space 58

3.5 Orthogonal projection and complement 66

3.6 Weak convergence 71

4 Operators on Hilbert Spaces 75

4.1 The adjoint of a bounded linear operator 75

4.2 Compact operators 82

5 Spectral Theory 89

5.1 The spectrum and the resolvent 89

5.2 Spectral theorem for compact self-adjoint operators 93.

Notes:

Includes bibliographical references (pages 129-130) and index.

ISBN:

9812565639

9812566864

OCLC:

67405711