1.3. Analytic functions 9

Contents

Introduction 2

Chapter 1. Background on infinite-dimensional holomorphy 3 1.1. Continuous polynomials and symmetric tensor products 3

1.2. Linear subspaces in zeros of polynomials 8

1.3. Analytic functions 9

1.4. The Aron-Berner extension 12

1.5. Concept of regularity 14

1.6. Hilbert-Schmidt polynomials 16

1.7. Reproducing kernels 20

Chapter 2. Descriptions of topological spectra and applications 22

2.1. Spectra of algebras of polynomials 23

2.2. Applications for symmetric polynomials 29

2.3. Polynomials on tensor products 31

2.4. The spectrum of H

(X) endowed with the Gelfand topology 34 2.5. The Gelfand transformation and linear structures on M

42

2.6. Linearity of topologies on spectra 47

2.7. Discontinuous complex homomorphisms and Michael’s problem 50

2.8. Continuous homomorphisms 51

2.9. Continuous derivations 53

2.10. Ball algebras of analytic functions 55

2.11. C

-algebras of continuous functions 57

Chapter 3. Hardy spaces associated with topological groups 59 3.1. Hardy spaces on compact infinite-dimensional group orbits 60 3.2. Symmetric Fock spaces associated with matrix unitary groups 74 3.3. Hardy spaces associated with infinite-dimensional unitary groups 86 3.4. Hardy spaces on irreducible orbits of locally compact groups 98 Chapter 4. Reproducing kernel spaces of analytic functions 111

4.1. Abstract Hardy spaces 111

4.2. Hilbert-Schmidt analytic functions 123

4.3. Holomorphicity and generalized symmetric Fock spaces 130

Bibliography 138

1