Classical Analysis on Normed Spaces

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Created on 26 May, 1995. Last Update: 25 March , 1997.

Preface

This book provides an elementary introduction to classical analysis on normed spaces with special attention to nonlinear topics such as fixed points, calculus and ordinary differential equations. It is for beginners who want to get through the basic material as soon as possible and then move on to do their own research immediately. It assumes only general knowledge in finite dimensional linear algebra, simple calculus and elementary complex analysis. Since the treatment is self-contained with sufficient details, even an undergraduate with mathematical maturity should have no trouble to work through it alone. Various chapters can be integrated into parts of a Master Degree Program by course work organized by any regional university. Restricted to finite dimensional spaces rather than normed spaces, selected chapters can be used for a coursein advanced calculus. Time will tell whether our vector-valued Taylor's formula and test for local extreme values will be supported and popularized into various undergraduate textbooks. We also hope that Engineers and Physicists would find this book to be a handy reference in classical analysis. High school teachers may be interested to enrich their programs by including the generalization of triangles and tetrahedra as treated in our chapter 4.

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Table of Contents

  1. Metric Spaces
    1. Standard Finite Dimensional Vector Spaces
    2. Convergent Sequences in Metric Spaces
    3. Continuous Maps
    4. Open Sets
    5. Closures of Sets
    6. Characterization of Continuity
    7. Duality of Closure-Interior Operators
    8. Partition of Unity
  2. Complete, Compact and Connected Sets
    1. Cauchy Sequences
    2. Bounded Sets
    3. Upper and Lower Limits
    4. Complete Sets
    5. Precompact Sets
    6. Compactness
    7. Continuous Maps on Compact Spaces
    8. Uniform Continuity
    9. Connected Sets
    10. Components
  3. Banach Spaces
    1. Uniform Convergence
    2. Bounded Continuous Functions
    3. Sequence Spaces
    4. Continuous Linear Maps
    5. Examples of Continuous Linear Maps
    6. Finite Dimensional Normed Spaces
    7. Infinite Dimensional Compact Sets
    8. Approximation in Function Spaces
  4. Simplicial Complexes
    1. Geometrically Independent Sets
    2. Convex Sets in Normed Spaces
    3. Simplexes
    4. Affine Maps
    5. Simplicial Complexes
    6. Small Simplexes
    7. Barycentric Subdivisions
    8. Simplicial Approximations
    9. Existence of Simplicial Approximations
    10. A Combinatorial Lemma with Application
  5. Topological Fixed Points
    1. Antipodal Maps
    2. Retracts and Fixed Points
    3. Fixed Points of Compact Maps
    4. Compact Fields and their Homotopies
    5. Extension Property
    6. Properties of Compact Fields in Normed Spaces
  6. Foundation of Functional Analysis
    1. Transfinite Induction
    2. Hahn-Banach Extension Theorems
    3. Extension of Continuous Linear Forms
    4. Closed Hyperplanes
    5. Separation by Hyperplanes
    6. Extreme Points
    7. Baire's Property
    8. Uniform Boundedness
    9. Open Map and Closed Graph Theorems
  7. Natural Constructions
    1. Bidual Spaces
    2. Quotient Spaces
    3. Duality of Subspaces and Quotients
    4. Direct Sums
    5. Transposes
    6. Reflexive Spaces
    7. Weak convergence
    8. Weak-Star Convergence
  8. Complex Analysis
    1. Derivatives of Vector Maps
    2. Integrals of Regulated Maps
    3. Fundamental Theorems of Calculus
    4. Holomorphic Maps of One Complex Variable
    5. Series Expansion
    6. Spectrum
    7. Spectral Radius
    8. Holomorphic Maps of an Operator
  9. Differentiation in Banach Spaces
    1. Differentiable Maps
    2. Mean-Value Theorem
    3. Partial Derivatives
    4. Fixed Points of Contractions
    5. Inverse and Implicit Mapping Theorems
    6. Local Properties of Differentiable Maps
  10. Polynomials and Higher Derivatives
    1. Multilinear Maps on Banach Spaces
    2. Polynomials on Banach Spaces
    3. Higher Derivatives
    4. Cn-Maps
    5. Taylor's Expansion
    6. Higher Chain Formula and Higher Product Formula
  11. Ordinary Differential Equations
    1. Local Existence and Uniqueness
    2. Integral Curves
    3. Linear Equations
    4. Exponential Functions of Matrices
    5. Global Dependence on Initial Conditions
    6. Solutions without Uniqueness
  12. Compact Linear Operators
    1. Basic Properties
    2. Riesz-Schauder Theory
    3. Spectrum of a Compact Operator
    4. Existence of Invariant Subspaces
  13. Operators on Hilbert Spaces
    1. Complex Inner Product Spaces
    2. Orthogonality in Inner Product Spaces
    3. Orthonormal Bases of Hilbert Spaces
    4. Orthogonal Complements
    5. Adjoints
    6. Quadratic Forms
    7. Normal Operators
    8. Self-Adjoint Operators
    9. Projectors and Closed Vector Subspaces
    10. Partial Order of Operators
    11. Eigenvalues
  14. Spectral Properties of Hilbert Spaces
    1. Spectrum of an Operator
    2. Approximate Spectrum
    3. Weak Convergence
    4. Diagonal Operators
    5. Compact Operators
    6. Functional Calculus of Self-Adjoint Operators
    7. Polar Decomposition
  15. Tensor Products
    1. Algebraic Tensor Products of Vector Spaces
    2. Tensor Products of Linear Maps
    3. Independent Sets in Tensor Products
    4. Matrix Representations
    5. Projective Norms on Tensor Products
    6. Inductive Norms
    7. Tensor Product of Hilbert Spaces
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American Mathematical Review--96h:00003

World Scientific Publishing Co., Inc., River Edge, NJ, 1995. xviii+356 pp. $74.00. ISBN 981-02-2137-1

The author aims to write "an elementary introduction to classical analysis on normed spaces", designed "for beginners who want to get through the basic material as soon as possible". This book "assumes only general knowledge in finite-dimensional linear algebra, simple calculus and elementary complex analysis". The reviewer feels that Ma has been successful in meeting this aim. The material is well organized and concisely (but fully) presented, with many examples and exercises. A modest amount of motivation is presented, but few historical remarks are made (although there is a list of 165 references for further reading).

There are 15 chapters. The first three deal with metric and normed spaces, completeness, compactness and connectedness, and an introduction to Banach spaces. Chapters 4 and 5 give the results from algebraic topology needed for fixed-point and related theorems. The next two chapters present the main results for Banach spaces, including a study of the bidual and weak and weak-star convergence. Chapter 8 discusses holomorphic vector-valued functions on open sets in C, including results concerning the spectrum of a matrix.

Chapters 9--11 discuss differentiation in Banach spaces, including the implicit function theorem and Lagrange multipliers (in Rn; polynomials, higher-order derivatives and Taylor's theorem; and ordinary differential equations of Banach-valued functions. In Chapter 12 the author presents the Riesz-Schauder theory for compact linear operators in a Banach space. In the next chapter he studies Hilbert spaces and in Chapter 14 he makes a study of normal and selfadjoint operators. While he does not develop the integral version of the spectral theorem, he does give a functional calculus for selfadjoint operators. The final chapter is an introduction to tensor products of spaces and operators.

Although the typography is not pleasing to the reviewer's eye, many students and instructors will welcome this book as an accessible introduction to these areas.

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Review by Math. Z. * 46003

Singapore: World Scientific, (ISBN 981-02-2137-1). xvi, 356 p. (1995).

This book provides an elementary but accelerated introduction to classical analysis on normed spaces with emphasis on nonlinear topics such as fixed points, calculus and ordinary differential equations. It assumes only general knowledge in finite-dimensional linear algebra, simple calculus and elementary complex analysis, since the rest of the treatment is self-contained. Almost each paragraph contains some illustrative exercises (not very challenging, needing no hints) and each chapter ends with references to informative titles. The general references at the end of the book counts 165 titles.

The first three chapters contains the necessary background for any course in analysis: topology (for metric spaces) and Banach spaces (including Ascoli and Stone-Weierstrass theorems).

Chapter 4 is devoted to simplicial complexes and simplicial approximations, opening the way to simplicial homology (not covered in this book).

Chapter 5 deals with topological fixed point theory, Brouwer's fixed point theorem being derived from the Borsuk-Ulam theorem. It ends with the theorem on invariance of domain.

The next two chapters contain standard topics of linear functional analysis, including Hahn-Banach and open mapping theorems. The weak topologies are not mentioned, weak convergence for sequences being used instead.

Vector valued maps of a scalar variable are introduced in chapter 8. The holomorphic functions are treated in the context of Banach spaces, with applications to the resolvent map.

The chapters 9, 10 are devoted to advanced calculus, including the implicit mapping theorem and Taylor's formula in Banach spaces.

Chapter 11 deals with the initial value problem x'=f(t,z). Linear ODEs are treated via exponentiation not requiring Jordan forms. A generalized Peano theorem is presented in infinite-dimensional spaces.

The next three chapters deal with compact operators, the Fredholm alternative, operators in Hilbert spaces, including polar decomposition.

The last chapter (the 15th) is devoted to tensor products with some improvements due to the author.

The book is a valuable tool for a consistent course for a master degree, being written in a very accurate style. V. Anisiu (Cluj-Napoca)

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Typographical Corrections

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Address of Publisher

wspc@scri.fsu.edu (US office)
wspc@wspc.demon.co.uk (UK office)
worldscp@singnet.com.sg (Singapore office)
Home Page of World Scientific Pub. Co.
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