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Functional Analysis and Operator Theory

Nicola Arcozzi

Abstract

Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of unitary, and of self-adjoint operators

Functional Analysis and Operator Theory

Abstract

Lecture note topics: 1. Some tools from real and complex analysis, 2. Hilbert spaces, 3. Banach spaces, 4. Compact operators and their spectra, 5. Intermezzo: reproducing kernel Hilbert spaces, 6. Banach algebras ,7. Spectral theory of unitary, and of self-adjoint operators
Paper Structure (141 sections, 201 theorems, 792 equations)

This paper contains 141 sections, 201 theorems, 792 equations.

Table of Contents

  1. Some tools from real and complex analysis
  2. The definition of Banach space
  3. Normed linear spaces
  4. Complete metric spaces and Banach spaces
  5. Borel measures
  6. Borel measures on the real line
  7. Borel measurable functions and the Lebesgue-Hausdorff theorem
  8. Fourier transforms and series
  9. The Fourier transform
  10. Fourier series
  11. Basic theory of harmonic functions
  12. The Dirichlet problem for the Laplace equation in the upper half plane
  13. The Fourier transforms of Borel measures
  14. Herglotz representation
  15. Bochner's theorem
  16. ...and 126 more sections

Key Result

Theorem 1.1

Let $(X, d)$ be a metric space. Then there exists a complete metric space $(\tilde{X}, \tilde{d})$, the completion of $(X, d)$, and an injective map $i \colon X \rightarrow \tilde{X}$ such that: The completion is unique in the following sense. For any other complete metric space $(Z, \delta)$ endowed with an injection $j \colon X \rightarrow Z$ satisfying properties (i) $\delta(j(x),j(y))=d(x,y)

Theorems & Definitions (386)

  • Theorem 1.1
  • proof
  • Proposition 1.1
  • proof
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.1
  • Lemma 1.1
  • proof
  • ...and 376 more