UID:
almahu_9947367855002882
Format:
1 online resource (469 p.)
ISBN:
1-281-79749-9
,
9786611797492
,
0-08-087175-5
Series Statement:
North-Holland mathematics studies ; 64
Content:
Differential Calculus and Holomorphy
Note:
Description based upon print version of record.
,
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
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1.0 Definition of differentiability in normed spaces1.1 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
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2.6 Series of homogeneous polynomials2.7 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)
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Chapter 5. Approximation and density results5.1 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
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Chapter 8. Nuclearity of spaces of holomorphic or C8 mappings8.1 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
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11.1 The Weïerstrass preparation theorem
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English
Additional Edition:
ISBN 0-444-86397-4
Language:
English
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