This book presents the fundamental function spaces and their duals, explores operator theory and finally develops the theory of distributions up to significant applications such as Sobolev spaces and Dirichlet problems. Includes an assortment of well formulated exercises, with answers and hints collected at the end of the book.
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Prologue: Sequences.- 1 Countability.- 2 Separability.- 3 The Diagonal
Procedure.- 4 Bounded Sequences of Continuous Linear Maps.- I Function Spaces
and Their Duals.- 1 The Space of Continuous Functions on a Compact Set.- 1
Generalities.- 2 The StoneWeierstrass Theorems.- 3 Ascolis Theorem.- 2
Locally Compact Spaces and Radon Measures.- 1 Locally Compact Spaces.- 2
Daniells Theorem.- 3 Positive Radon Measures.- 3A Positive Radon Measures on
$${{\mathbb{R}}^{d}}
$$ and the Stieltjes Integral.- 3B Surface Measure on Spheres in $${{\mathbb{R}}^{d}}
$$.- 4 Real and Complex Radon Measures.- 3 Hilbert Spaces.- 1 Definitions,
Elementary Properties, Examples.- 2 The Projection Theorem.- 3 The Riesz
Representation Theorem.- 3A Continuous Linear Operators on a Hilbert Space.-
3B Weak Convergence in a Hilbert Space.- 4 Hilbert Bases.- 4 LpSpaces.- 1
Definitions and General Properties.- 2 Duality.- 3 Convolution.- II
Operators.- 5 Spectra.- 1 Operators on Banach Spaces.- 2 Operators in Hilbert
Spaces.- 2A Spectral Properties of Hermitian Operators.- 2B Operational
Calculus on Hermitian Operators.- 6 Compact Operators.- 1 General
Properties.- lA Spectral Properties of Compact Operators.- 2 Compact
Selfadjoint Operators.- 2A Operational Calculus and the Fredholm Equation.-
2B Kernel Operators.- III Distributions.- 7 Definitions and Examples.- 1 Test
Functions.- lA Notation.- 1B Convergence in Function Spaces.- 1C Smoothing.-
1D C?Partitions of Unity.- 2 Distributions.- 2A Definitions.- 2B First
Examples.- 2C Restriction and Extension of a Distribution to an Open Set.- 2D
Convergence of Sequences of Distributions.- 2E Principal Values.- 2F Finite
Parts.- 3 Complements.- 3A Distributions of Finite Order.- 3B The Support of
a Distribution.- 3C Distributions with Compact Support.- 8 Multiplication and
Differentiation.- 1 Multiplication.- 2 Differentiation.- 3 Fundamental
Solutions of a Differential Operator.- 3A The Laplacian.- 3B The Heat
Operator.- 3C The Cauchy-Riemann Operator.- 9 Convolution of Distributions.-
1 Tensor Product of Distributions.- 2 Convolution of Distributions.- 2A
Convolution in ??.- 2B Convolution in D?.- 2C Convolution of a Distribution
with a Function.- 3 Applications.- 3A Primitives and Sobolevs Theorem.- 3B
Regularity.- 3C Fundamental Solutions and Partial Differential Equations.- 3D
The Algebra D+?.- 10 The Laplacian on an Open Set.- 1 The spaces H1(?) and
H01(?).- 2 The Dirichlet Problem.- 2A The Dirichlet Problem.- 2B The Heat
Problem.- 2C The Wave Problem.- Answers to the Exercises.
