Differential Calculus and Holomorphy

Name: Differential Calculus and Holomorphy | Real and Complex Analysis in Locally Convex Spaces
Brand: North-Holland
Price: 195.0 EUR
Availability: OnlineOnly

Real and Complex Analysis in Locally Convex Spaces

Colombeau J. F. Colombeau(Autor*in)

North-Holland (Verlag)

Erschienen am 14. Mai 2014

468 Seiten

E-Book

PDF mit Wasserzeichen-DRM

Systemvoraussetzungen

978-0-08-087175-2 (ISBN)

195,00 €inkl. 7% MwSt.

Systemvoraussetzungen

für PDF mit Wasserzeichen-DRM

E-Book Einzellizenz

Als Download verfügbar

Beschreibung

Weitere Details

Weitere Ausgaben

Inhalt

Front Cover
Differential Calculus and Holomorphy
Copyright Page
Contents
General Introduction
Part 0: A Review of the Linear Background
Introduction
0.1 Locally convex spaces
0.2 Bornological vector spaces
0.3 Elements of duality
0.4 Compact and nuclear mappings in normed spaces
0.5 Schwartz and nuclear spaces
0.6 A few classes of infinite dimensional spaces
0.7 Compact and nuclear subsets of Fréchet spaces
0.8 Multilinear mappings and polynomials
Part I: Basic Differential Calculus and Holomorphy
Introduction
Chapter 1. Differentiable mappings, basic properties
1.0 Definition of differentiability in normed spaces
1.1 Definition of (Silva) differentiable mappings
1.2 Definitions of Cn and C8 mappings
1.3 Mean value theorem and Taylor's formulas
1.4 C8 mappings in the enlarged sense
1.5 Mappings which are "locally differentiable between normed spaces
1.6 C8 mappings of uniform bounded type
Chapter 2. Holomorphic mappings, basic properties
2.1 Gateaux analytic mappings
2.2 Silva holomorphic mappings
2.3 Holomorphic mappings and Silva holomorphic mappings in the enlarged sense
2.4 C8 differentiability of holomorphic mappings
2.5 An example
2.6 Series of homogeneous polynomials
2.7 Holomorphic mappings of uniform bounded type
2.8 Holomorphic representation of Fock spaces of Boson Fields
Chapter 3. Classical properties of holomorphic mappings
3.1 Vector valued holomorphy versus scalar valued holomorphy
3.2 Zorn's theorem
3.3 Hartogs' theorem
3.4 Montel's theorem
Chapter 4. Topologies on E(O, F) and HS (O, F)
4.1 Natural topologies on E(O, F) and HS(O, F)
4.2 Completeness of HS(O, F) and E(O,F)
4.3 Schwartz property of HS (O, F) and E(O, F)
4.4 Reflexivity of HS(O, F) and E(O,F)
Chapter 5. Approximation and density results
5.1 A density result in HS (O, F)
5.2 A density result in E(O,F)
5.3 c8 partitions of unity
Chapter 6. ?-product and kernel theorems
6.1 Schwartz ?-product in spaces of holomorphic functions
6.2 Schwartz e-product in spaces of C8 function
6.3 Approximation property in Hs(O) and E(O)
Chapter 7. The Fourier-Borel and Fourier transforms
7.1 Preliminary results on the Fourier-Borel transform
7.2 The Fourier-Borel isomorphism
7.3 Holomorphic germs
7.4 The Fourier transform and the Paley-Wiener- Schwartz isomorphisms
Chapter 8. Nuclearity of spaces of holomorphic or C8 mappings
8.1 Nuclearity of HS(O)
8.2 Strong nuclearity of HS(O)
8.3 Non nuclearity of E(O)
Part II: Convolution and ? Equations
Introduction
Chapter 9. Convolution equations in P(E)
9.1 Formal power series and duality
9.2 A division result
9.3 The convolution operators on P(E)
9.4 Existence of solutions
Chapter 10. Convolution equations in spaces of entire functions of exponential type
10.1 The convolution operators on JH's(O)
10.2 Approximation of the solutions
10.3 Existence of solutions
Chapter 11. Division of distributions
11.1 The Weïerstrass preparation theorem
11.2 Division by a complex polynomial
11.3 Division of distributions by holomorphic functions
11.4 Application to existence of solutions
11.5 Division by real analytic functions of finite type
11.6 Impossibility of the division by real polynomials
Chapter 12. Convolution equations in spaces of holomorphic functions
12.1 The convolution operators on HS(E)
12.2 Entire functions of nuclear bounded type on a Banach space
12.3 The convolution operators on Hu, b(E)
12.4 The convolution operators on HN, b(E)
12.5 A division result
12.6 Existence and approximation results in HN, b(E)
12.7 Existence and approximation results in HN, b(E) and HS(E)
12.8 Convolution operators of finite type
Chapter 13. Linear finite difference partial differential equations in E(E)
13.1 A division result by imaginary exponential polynomials
13.2 A Paley-Wiener-Schwartz theorem and a division result
13.3 Existence and approximation of solutions in normed spaces
13.4 Existence of solutions in locally convex spaces
Chapter 14. Pseudo-convex domains and approximation results
14.1 Glimpse at pseudo-convexity and domains of holomorphy
14.2 The Lévi problem
14.3 The Runge approximation theorem
14.4 An approximation theorem
Chapter 15. The ? equation
15.1 Differential forms and ? operator
15.2 A review of Hormander's L2 estimates and existence theorems
15.3 A review on integration in Hilbert spaces
15.4 A basic existence result
15.5 An hypoellipticity result
15.6 The ? equation in a scale of Hilbert spaces
15.7 Existence of C8 solutions in pseudo-convex open subsets of DFN space
15.8 Existence of C8 solutions in nuclear l . c . s.
Chapter 16. Some applications of the ? equation
16.1 Solution of the first Cousin problem
16.2 A counterexample
16.3 On solutions of some homogeneous convolution equations
Bibliographic Notes
Bibliography
Index

Systemvoraussetzungen

Als PDF speichern Als Link merken