Derived Functors in Functional Analysis

ISBN-13: 9783540002369 / Angielski / Miękka / 2002 / 138 str.

Olaf Steinbach; J. Wengenroth; Jochen Wengenroth

Opis

The text contains for the first time in book form the state of the art of homological methods in functional analysis like characterizations of the vanishing of the derived projective limit functor or the functors Ext¹ (E, F) for Frechet and more general spaces. The researcher in real and complex analysis finds powerful tools to solve surjectivity problems e.g. on spaces of distributions or to characterize the existence of solution operators.
The requirements from homological algebra are minimized: all one needs is summarized on a few pages. The answers to several questions of V.P. Palamodov who invented homological methods in analysis also show the limits of the program.

Spis treści

1 Introduction 1

2 Notions from homological algebra 7

2.1 Derived Functors 7

2.2 The category of locally convex spaces 13

3 The projective limit functor for countable spectra 17

3.1 Projective limits of linear spaces 17

3.2 The Mittag-Leffler procedure 23

3.3 Projective limits of locally convex spaces 38

3.4 Some Applications 50

3.4.1 The Mittag-Leffler theorem 50

3.4.2 Separating singularities 51

3.4.3 Surjectivity of the Cauchy-Riemann operator 51

3.4.4 Surjectivity of P(D) on spaces of smooth functions 52

3.4.5 Surjectivity of P(D) the space of distributions 52

3.4.6 Differential operators for ultradifferentiable functions of Roumieu type 54

4 Uncountable projective spectra 59

4.1 Projective spectra of linear spaces 59

4.2 Insertion: The completion functor 68

4.3 Projective spectra of locally convex spaces 70

5 The derived functors of Hom 77

5.1 Ext^k in the category of locally convex spaces 77

5.2 Splitting theory for Fréchet spaces 86

5.3 Splitting in the category of (PLS)-spaces 96

6 Inductive spectra of locally convex spaces 109

7 The duality functor 119

References 129

Index