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Functional analysis : introduction to further topics in analysis /
"This is the fourth and final volume in the Princeton Lectures in Analysis, a series of textbooks that aim to present, in an integrated manner, the core areas of analysis. Beginning with the basic facts of functional analysis, this volume looks at Banach spaces, Lp spaces, and distribution theo...
| Clasificación: | Libro Electrónico |
|---|---|
| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Princeton, N.J. :
Princeton University Press,
2011.
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| Colección: | Princeton lectures in analysis ;
4. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Contents; Foreword; Preface; Chapter 1. L[sup(p)] Spaces and Banach Spaces; 1 L[sup(p)] spaces; 1.1 The Hölder and Minkowski inequalities; 1.2 Completeness of L[sup(p)]; 1.3 Further remarks; 2 The case p = ∞; 3 Banach spaces; 3.1 Examples; 3.2 Linear functionals and the dual of a Banach space; 4 The dual space of L[sup(p)] when 1 ≤ p < ∞; 5 More about linear functionals; 5.1 Separation of convex sets; 5.2 The Hahn-Banach Theorem; 5.3 Some consequences; 5.4 The problem of measure; 6 Complex L[sup(p)] and Banach spaces; 7 Appendix: The dual of C(X)
- 7.1 The case of positive linear functionals7.2 The main result; 7.3 An extension; 8 Exercises; 9 Problems; Chapter 2. L[sup(p)] Spaces in Harmonic Analysis; 1 Early Motivations; 2 The Riesz interpolation theorem; 2.1 Some examples; 3 The L[sup(p) theory of the Hilbert transform; 3.1 The L[sup(2)] formalism; 3.2 The L[sup(p) theorem; 3.3 Proof of Theorem 3.2; 4 The maximal function and weak-type estimates; 4.1 The L[sup(p)] inequality; 5 The Hardy space H[sup(1)][sub(r); 5.1 Atomic decomposition of H[sup(1)[sub(r); 5.2 An alternative definition of H[sup(1)[sub(r)]
- 5.3 Application to the Hilbert transform6 The space H[sup(1)[sub(r)] and maximal functions; 6.1 The space BMO; 7 Exercises; 8 Problems; Chapter 3. Distributions: Generalized Functions; 1 Elementary properties; 1.1 Definitions; 1.2 Operations on distributions; 1.3 Supports of distributions; 1.4 Tempered distributions; 1.5 Fourier transform; 1.6 Distributions with point supports; 2 Important examples of distributions; 2.1 The Hilbert transform and pv(1/x); 2.2 Homogeneous distributions; 2.3 Fundamental solutions
- 2.4 Fundamental solution to general partial differential equations with constant coefficients2.5 Parametrices and regularity for elliptic equations; 3 Caldeórn-Zygmund distributions and L[sup(p)] estimates; 3.1 Defining properties; 3.2 The L[sup(p) theory; 4 Exercises; 5 Problems; Chapter 4. Applications of the Baire Category Theorem; 1 The Baire category theorem; 1.1 Continuity of the limit of a sequence of continuous functions; 1.2 Continuous functions that are nowhere differentiable; 2 The uniform boundedness principle; 2.1 Divergence of Fourier series; 3 The open mapping theorem
- 3.1 Decay of Fourier coefficients of L[sup(1)]-functions4 The closed graph theorem; 4.1 Grothendieck's theorem on closed subspaces of L[sup(p)]; 5 Besicovitch sets; 6 Exercises; 7 Problems; Chapter 5. Rudiments of Probability Theory; 1 Bernoulli trials; 1.1 Coin flips; 1.2 The case N = ∞; 1.3 Behavior of S[sub(N)] as N → ∞, first resultes; 1.4 Central limit theorem; 1.5 Statement and proof of the theorem; 1.6 Random series; 1.7 Random Fourier series; 1.8 Bernoulli trials; 2 Sums of independent random variables; 2.1 Law of large numbers and ergodic theorem; 2.2 The role of martingales