From Euclidean to Hilbert Spaces: Introduction to Functional Analysis and its Applications

✍ Scribed by Edoardo Provenzi

• Publisher: Wiley-ISTE
• Year: 2021
• Tongue: English
• Leaves: 355
• Edition: 1
• Category: Library

✦ Synopsis

From Euclidian to Hilbert Spaces analyzes the transition from finite dimensional Euclidian spaces to infinite-dimensional Hilbert spaces, a notion that can sometimes be difficult for non-specialists to grasp. The focus is on the parallels and differences between the properties of the finite and infinite dimensions, noting the fundamental importance of coherence between the algebraic and topological structure, which makes Hilbert spaces the infinite-dimensional objects most closely related to Euclidian spaces.

The common thread of this book is the Fourier transform, which is examined starting from the discrete Fourier transform (DFT), along with its applications in signal and image processing, passing through the Fourier series and finishing with the use of the Fourier transform to solve differential equations.

The geometric structure of Hilbert spaces and the most significant properties of bounded linear operators in these spaces are also covered extensively. The theorems are presented with detailed proofs as well as meticulously explained exercises and solutions, with the aim of illustrating the variety of applications of the theoretical results.

✦ Table of Contents

Cover
Half-Title Page
Dedication
Title Page
Copyright Page
Contents
Preface
Chapter 1. Inner Product Spaces (Pre-Hilbert)
1.1. Real and complex inner products
1.2. The norm associated with an inner product and normed vector spaces
1.2.1. The parallelogram law and the polarization formula
1.3. Orthogonal and orthonormal families in inner product spaces
1.4. Generalized Pythagorean theorem
1.5. Orthogonality and linear independence
1.6. Orthogonal projection in inner product spaces
1.7. Existence of an orthonormal basis: the Gram-Schmidt process
1.8. Fundamental properties of orthonormal and orthogonal bases
1.9. Summary
Chapter 2. The Discrete Fourier Transform and its Applications to Signal and Image Processing
2.1. The space l2(ZN) and its canonical basis
2.1.1. The orthogonal basis of complex exponentials in l2 (ZN)
2.2. The orthonormal Fourier basis of l2 (ZN)
2.3. The orthogonal Fourier basis of
2.4. Fourier coefficients and the discrete Fourier transform
2.4.1. The inverse discrete Fourier transform
2.4.2. Definition of the DFT and the IDFT with the orthonormal Fourier basis
2.4.3. The real (orthonormal) Fourier basis
2.5. Matrix interpretation of the DFT and the IDFT
2.5.1. The fast Fourier transform
2.6. The Fourier transform in signal processing
2.6.1. Synthesis formula for 1D signals: decomposition on the harmonic basis
2.6.2. Signification of Fourier coefficients and spectrums of a 1D signal
2.6.3. The synthesis formula and Fourier coefficients of the unit pulse
2.6.4. High and low frequencies in the synthesis formula
2.6.5. Signal filtering in frequency representation
2.6.6. The multiplication operator and its diagonal matrix representation
2.6.7. The Fourier multiplier operator
2.7. Properties of the DFT
2.7.1. Periodicity of ẑ and ž
2.7.2. DFT and shift
2.7.3. DFT and conjugation
2.7.4. DFT and convolution
2.8. The DFT and stationary operators
2.8.1. The DFT and the diagonalization of stationary operators
2.8.2. Circulant matrices
2.8.3. Exhaustive characterization of stationary operators
2.8.4. High-pass, low-pass and band-pass filters
2.8.5. Characterizing stationary operators using shift operators
2.8.6. Frequency analysis of first and second derivation operators (discrete case)
2.9. The two-dimensional discrete Fourier transform (2D DFT)
2.9.1. Matrix representation of the 2D DFT: Kronecker product versus iteration of two 1D DFTs
2.9.2. Properties of the 2D DFT
2.9.3. 2D DFT and stationary operators
2.9.4. Gradient and Laplace operators and their action on digital images
2.9.5. Visualization of the amplitude spectrum in 2D
2.9.6. Filtering: an example of digital image filtering in a Fourier space
2.10. Summary
Chapter 3. Lebesgue’s Measure and Integration Theory
3.1. Riemann versus Lebesgue
3.2. σ-algebra, measurable space, measures and measured spaces
3.3. Measurable functions and almost-everywhere properties (a.e)
3.4. Integrable functions and Lebesgue integrals
3.5. Characterization of the Lebesgue measure on R and sets with a null Lebesgue measure
3.6. Three theorems for limit operations in integration theory
3.7. Summary
Chapter 4. Banach Spaces and Hilbert Spaces
4.1. Metric topology of inner product spaces
4.2. Continuity of fundamental operations in inner product spaces
4.2.1. Equivalence of separated topologies in finite-dimension vector spaces
4.3. Cauchy sequences and completeness: Banach and Hilbert
4.3.1. Completeness of vector spaces
4.3.2. Characterizing the completeness of normed vector spaces using series
4.3.3. Banach fixed-point theorem
4.4. Remarkable examples of Banach and Hilbert spaces
4.4.1. Lp and lp spaces: presentation and completeness
4.4.2. L∞ and l∞ spaces
4.4.3. Inclusion relationships between lp spaces
4.4.4. Inclusion relationships between Lp spaces
4.4.5. Density theorems in Lp(X,A,μ)
4.5. Summary
Chapter 5. The Geometric Structure of Hilbert Spaces
5.1. The orthogonal complement in a Hilbert space and its properties
5.2. Projection onto closed convex sets: theorem and consequences
5.2.1. Characterization of closed vector subspaces in Hilbert spaces
5.3. Polar and bipolar subsets of a Hilbert space
5.4. The (orthogonal) projection theorem in a Hilbert space
5.5. Orthonormal systems and Hilbert bases
5.5.1. Bessel’s inequality and Fourier coefficients
5.5.2. The Fischer-Riesz theorem
5.5.3. Characterizations of a Hilbert basis (or complete orthonormal system)
5.5.4. Isomorphisms between Hilbert spaces
5.5.5. l2(N,K) as the prototype of separable Hilbert spaces of infinite dimension
5.6. The Fourier Hilbert basis in L2
5.6.1. L2[-π, π] or L2[0, 2π]
5.6.2. L2pTq
5.6.3. L2[a,b]
5.6.4. Real Fourier series
5.6.5. Pointwise convergence of the real Fourier series: Dirichlet’s theorem
5.6.6. The Gibbs phenomenon and Cesàro summation
5.6.7. Speed of convergence to 0 of Fourier coefficients
5.6.8. Fourier transform in L2 (T) and shift
5.7. Summary
Chapter 6. Bounded Linear Operators in Hilbert Spaces
6.1. Fundamental properties of bounded linear operators between normed vector spaces
6.1.1. Continuity of linear operators defined on a finite-dimensional normed vector space
6.2. The operator norm, convergence of operator sequences and Banach algebras
6.2.1. A classical example of a non-bounded linear operator on a vector space of infinite dimension
6.3. Invertibility of linear operators
6.4. The dual of a Hilbert space and the Riesz representation theorem
6.4.1. The scalar product induced on the dual of a Hilbert space
6.5. Bilinear forms, sesquilinear forms and associated quadratic forms
6.5.1. The Lax-Milgram theorem and its consequences
6.6. The adjoint operator: presentation and properties
6.7. Orthogonal projection operators in a Hilbert space
6.7.1. Bounded multiplication operators and their relation to orthogonal projectors
6.7.2. Geometric realization of orthogonal projection operators via orthonormal systems
6.8. Isometric and unitary operators
6.8.1. Characterizations of isometric and unitary operators
6.8.2. Relationship between isometric and unitary operators and orthonormal systems
6.9. The Fourier transform on S(Rn), L1(Rn) and L2 (Rn)
6.9.1. The invariance of the Schwartz space with respect to the Fourier transform
6.9.2. Extension of the Fourier transform of S(Rn) to L1(Rn): the Riemann-Lebesgue theorem
6.9.3. Extension of the Fourier transform to a unitary operator on L2(Rn): the Fourier-Plancherel transform
6.9.4. Relationship between the Fourier-Plancherel transform and the Hermitian Hilbert basis
6.9.5. The Fourier transform and convolution
6.9.6. Convolution and Fourier transforms in L2: localization of the Fourier transform
6.10. The Nyquist-Shannon sampling theorem
6.10.1. The Nyquist frequency: aliasing and oversampling
6.11. Application of the Fourier transform to solve ordinary and partial differential equations
6.11.1. Solving an ordinary differential equation using the Fourier transform
6.11.2. The Fourier transform and partial differential equations
6.11.3. Solving the partial differential equation for heat propagation using the Fourier transform
6.12. Summary
Appendix 1. Quotient Space
Appendix 2. The Transpose (or Dual) of a Linear Operator
Appendix 3. Uniform, Strong and Weak Convergence
References
Index
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