15: Appendix B- Hilbert Spaces Overview
- Page ID
- 22941
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- 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.