Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

15: Appendix B- Hilbert Spaces Overview

Last updated

Feb 23, 2021
Save as PDF
- 14.5: Eigenfunctions of LTI Systems
- 15.1: Fields and Complex Numbers

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

15.1: Fields and Complex Numbers
This page discusses vector spaces in linear algebra and introduces fields, which are sets with addition and multiplication operations. Common fields include real and complex numbers. Complex numbers are explored in both Cartesian and polar forms, emphasizing their geometric interpretations. The text explains calculations of their magnitude and angle using Euler's Formula, and covers arithmetic operations, highlighting the efficiency of polar coordinates for vector multiplication and operations.
15.2: Vector Spaces
This page discusses vector spaces as collections of vectors defined by vector addition and scalar multiplication, with scalars from a specified field. It distinguishes between real and complex vector spaces and outlines key properties such as closure, commutativity, and the existence of a zero vector. Examples include $L^p$ spaces and $\mathbb{R}^n$ , while noting that certain sets like $\mathbb{R}_+^2$ and $D$ do not meet vector space criteria.
15.3: Norms
This page introduces norms as functions measuring the size of vectors, highlighting properties such as positivity, scaling, and adherence to the triangle inequality. It provides examples including $\ell^1$ , $\ell^2$ , and $\ell^{\infty}$ in $\mathbb{R}^n$ and extends the concept to discrete time signals, sequences, and continuous functions. These norms serve to quantify the size of elements across different mathematical contexts.
15.4: Inner Products
This page explains the inner product (or dot product) for vectors in $\mathbb{R}^n$ , detailing its calculation as the sum of component products and its geometric interpretation as the strength of one vector in the direction of another. It also notes that an inner product of zero indicates orthogonality and outlines key properties such as conjugate symmetry, scaling, and additivity, emphasizing its importance in vector space analysis.
15.5: Hilbert Spaces
This page explains Hilbert spaces as complete inner product spaces that facilitate the study of mathematical concepts like Fourier expansion and quantum mechanics. Named after David Hilbert, they employ an inner product to define a norm. Examples include $\mathbb{C}^n$ , $L^{2}(\mathbb{R})$ , and $\ell^{2}(\mathbb{Z})$ . Hilbert spaces play a crucial role in functional analysis.
15.6: Cauchy-Schwarz Inequality
This page emphasizes the significance of the Cauchy-Schwarz inequality in linear algebra and signal processing, particularly in matched filters for pattern detection. It explains the inequality's role in maximizing inner products of normalized vectors and its implications for signal correlation. The discussion includes the implementation of matched filters through normalized cross-correlation and convolution, with an emphasis on the Fast Fourier Transform for efficiency.
15.7: Common Hilbert Spaces
This page discusses four essential Hilbert spaces for signal and system analysis: $\mathbb{R}^n$ , $\mathbb{C}^n$ , $L^2([a,b])$ , and $\ell^{2}(\mathbb{Z})$ , each with unique inner products suited for different signals. It emphasizes their role in Fourier analysis and notes that only $L^2(\mathbb{R})$ qualifies as a Hilbert space among the $L^p(\mathbb{R})$ spaces.
15.8: Types of Bases
This page discusses normalized, orthogonal, and orthonormal bases in vector spaces, emphasizing their properties and utility, particularly in $\mathbb{R}^2$ . It illustrates how vectors can be represented more simply using orthonormal bases, aiding in the calculation of coefficients in linear combinations. Additionally, it explains orthonormal basis expansions in Hilbert spaces, where any vector can be expressed as a sum of basis vectors weighted by coefficients derived from the inner product.
15.9: Orthonormal Basis Expansions
This page explores the decomposition of signals into smaller components using orthonormal bases in the context of eigenvectors and LTI systems. It defines a basis in vector spaces, introduces expansion coefficients, and provides examples in $\mathbb{R}^2$ .
15.10: Function Space
This page discusses finding basis vectors for vector spaces, specifically focusing on the vector space $P_n$ of $n$ -th order polynomials. It presents examples of bases for $P_2$ and shows how any quadratic polynomial can be expressed as a linear combination of these bases. The text also introduces infinite-dimensional spaces, exemplified by the basis $e^{j \omega_{0} n t}$ for $L^2([0,T])$ , highlighting the complexities and similarities between finite and infinite-dimensional spaces.
15.11: Haar Wavelet Basis
This page discusses Fourier series and wavelets as bases for $L^2([0,T])$ , highlighting the limitations of Fourier series, particularly in image processing due to Gibbs phenomena. It emphasizes the advantages of wavelets, which are local in time, and details the Haar wavelet basis, known for its effective representation and energy conservation.
15.12: Orthonormal Bases in Real and Complex Spaces
This page covers matrix transposition, inner products, and orthonormal bases in linear algebra. It defines the transpose operator, inner products for real/complex spaces, and the Hermitian transpose. The text discusses properties of orthonormal bases, illustrating how to express a vector as a linear combination of basis vectors and simplify coefficient calculations with the relation $\boldsymbol{\alpha} = B^H \boldsymbol{x}$ for orthonormal bases.
15.13: Plancharel and Parseval's Theorems
This page explains Parseval's and Plancherel theorems, which connect time-domain signal energy to frequency-domain coefficients. Parseval's theorem equates a signal's energy to the sum of squares of its Fourier series coefficients. Plancherel's theorem maintains that the inner product of two signals is unchanged in their coefficient space, linking inner products in L² spaces to their Fourier representations.
15.14: Approximation and Projections in Hilbert Space
This page explains finding the closest point on a line to a point in a plane through vector projections. It details projecting a vector onto a subspace using an orthonormal basis, highlighting examples in $\mathbb{R}^3$ and signal space approximations. Key concepts include orthogonality in projections and applications in higher dimensions, particularly in Fourier series reconstruction.

Support Center

How can we help?