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Engineering LibreTexts

15: Appendix B- Hilbert Spaces Overview

  • Page ID
    • 15.1: Fields and Complex Numbers
      An introduction to fields and complex numbers.
    • 15.2: Vector Spaces
      This module will define what a vector space is and provide useful examples to the reader.
    • 15.3: Norms
      This module will define a norm and give examples and properties of it.
    • 15.4: Inner Products
      This module describes the concept of inner products, which leads into our introduction of Hilbert spaces. Examples and properties of both of these concepts are discussed.
    • 15.5: Hilbert Spaces
      This module will provide an introduction to the concepts of Hilbert spaces.
    • 15.6: Cauchy-Schwarz Inequality
      This module provides both statement and proof of the Cauchy-Schwarz inequality and discusses its practical implications with regard to the matched filter detector.
    • 15.7: Common Hilbert Spaces
      This module will give an overview of the most common Hilbert spaces and their basic properties.
    • 15.8: Types of Bases
      This module discusses the different types of basis that leads up to the definition of an orthonormal basis. Examples are given and the useful of the orthonormal basis is discussed.
    • 15.9: Orthonormal Basis Expansions
      The module looks at decomposing signals through orthonormal basis expansion to provide an alternative representation. The module presents many examples of solving these problems and looks at them in several spaces and dimensions.
    • 15.10: Function Space
      This module gives an example on function space.
    • 15.11: Haar Wavelet Basis
      This module gives an overview of wavelets and their usefulness as a basis in image processing. In particular we look at the properties of the Haar wavelet basis.
    • 15.12: Orthonormal Bases in Real and Complex Spaces
      This module defines the terms transpose, inner product, and Hermitian transpose and their use in finding an orthonormal basis.
    • 15.13: Plancharel and Parseval's Theorems
      This module contains the definition of the Plancharel theorem and Parseval's theorem along with proofs and examples.
    • 15.14: Approximation and Projections in Hilbert Space
      This module introduces approximation and projections in Hilbert space.