From Euclidean to Hilbert spaces : introduction to functional analysis and its applications /: introduction to functional analysis and its applications. (2021)

• Record Type: Book
• Further Information: Note: Edoardo Provenzi.
• Authors: Provenzi, Edoardo
• Contents: Preface xi Chapter 1. Inner Product Spaces (Pre-Hilbert) 1 1.1. Real and complex inner products 1 1.2. The norm associated with an inner product and normed vector spaces 6 1.2.1. The parallelogram law and the polarization formula 9 1.3. Orthogonal and orthonormal families in inner product spaces 11 1.4. Generalized Pythagorean theorem 11 1.5. Orthogonality and linear independence 13 1.6. Orthogonal projection in inner product spaces 15 1.7. Existence of an orthonormal basis: the Gram-Schmidt process 19 1.8. Fundamental properties of orthonormal and orthogonal bases 20 1.9. Summary 28 Chapter 2. The Discrete Fourier Transform and its Applications to Signal and Image Processing 31 2.1. The space l 2 (ZN ) and its canonical basis 31 2.1.1. The orthogonal basis of complex exponentials in l 2 (ZN ) 34 2.2. The orthonormal Fourier basis of l 2 (ZN ) 38 2.3. The orthogonal Fourier basis of l 2 (ZN ) 40 2.4. Fourier coefficients and the discrete Fourier transform 41 2.4.1. The inverse discrete Fourier transform 44 2.4.2. Definition of the DFT and the IDFT with the orthonormal Fourier basis 46 2.4.3. The real (orthonormal) Fourier basis 47 2.5. Matrix interpretation of the DFT and the IDFT 48 2.5.1. The fast Fourier transform 51 2.6. The Fourier transform in signal processing 51 2.6.1. Synthesis formula for 1D signals: decomposition on the harmonic basis 51 2.6.2. Signification of Fourier coefficients and spectrums of a 1D signal 53 2.6.3. The synthesis formula and Fourier coefficients of the unit pulse 54 2.6.4. High and low frequencies in the synthesis formula 55 2.6.5. Signal filtering in frequency representation 59 2.6.6. The multiplication operator and its diagonal matrix representation 60 2.6.7. The Fourier multiplier operator 60 2.7. Properties of the DFT 61 2.7.1. Periodicity of ẑ and ? 62 2.7.2. DFT and shift 63 2.7.3. DFT and conjugation 67 2.7.4. DFT and convolution 68 2.8. The DFT and stationary operators 73 2.8.1. The DFT and the diagonalization of stationary operators 75 2.8.2. Circulant matrices 77 2.8.3. Exhaustive characterization of stationary operators 78 2.8.4. High-pass, low-pass and band-pass filters 82 2.8.5. Characterizing stationary operators using shift operators 83 2.8.6. Frequency analysis of first and second derivation operators (discrete case) 84 2.9. The two-dimensional discrete Fourier transform (2D DFT) 88 2.9.1. Matrix representation of the 2D DFT: Kronecker product versus iteration of two 1D DFTs 91 2.9.2. Properties of the 2D DFT 93 2.9.3. 2D DFT and stationary operators 95 2.9.4. Gradient and Laplace operators and their action on digital images 97 2.9.5. Visualization of the amplitude spectrum in 2D 97 2.9.6. Filtering: an example of digital image filtering in a Fourier space 100 2.10. Summary 102 Chapter 3. Lebesgue’s Measure and Integration Theory 105 3.1. Riemann versus Lebesgue 105 3.2. σ-algebra, measurable space, measures and measured spaces 106 3.3. Measurable functions and almost-everywhere properties (a.e) 108 3.4. Integrable functions and Lebesgue integrals 109 3.5. Characterization of the Lebesgue measure on ℝ and sets with a null Lebesgue measure 111 3.6. Three theorems for limit operations in integration theory 113 3.7. Summary 114 Chapter 4. Banach Spaces and Hilbert Spaces 115 4.1. Metric topology of inner product spaces 116 4.2. Continuity of fundamental operations in inner product spaces 120 4.2.1. Equivalence of separated topologies in finite-dimension vector spaces 128 4.3. Cauchy sequences and completeness: Banach and Hilbert 129 4.3.1. Completeness of vector spaces 133 4.3.2. Characterizing the completeness of normed vector spaces using series 135 4.3.3. Banach fixed-point theorem 139 4.4. Remarkable examples of Banach and Hilbert spaces 145 4.4.2. L ꚙ and l ꚙ spaces 156 4.4.3. Inclusion relationships between lp spaces 161 4.4.4. Inclusion relationships between Lp spaces 163 4.4.5. Density theorems in Lp (X, A, μ ) 165 4.5. Summary 169 Chapter 5. The Geometric Structure of Hilbert Spaces 171 5.1. The orthogonal complement in a Hilbert space and its properties 171 5.2. Projection onto closed convex sets: theorem and consequences 174 5.2.1. Characterization of closed vector subspaces in Hilbert spaces 180 5.3. Polar and bipolar subsets of a Hilbert space 182 5.4. The (orthogonal) projection theorem in a Hilbert space 185 5.5. Orthonormal systems and Hilbert bases 188 5.5.1. Bessel’s inequality and Fourier coefficients 189 5.5.2. The Fischer-Riesz theorem 192 5.5.3. Characterizations of a Hilbert basis (or complete orthonormal system) 194 5.5.4. Isomorphisms between Hilbert spaces 199 5.5.5. l 2 (N, K) as the prototype of separable Hilbert spaces of infinite dimension 201 5.6. The Fourier Hilbert basis in L 2 202 5.6.1. L 2 [-π, π ] or L2 [0, 2π ] 202 5.6.2. L 2 (T) 204 5.6.3. L 2 [a, b ] 205 5.6.4. Real Fourier series 206 5.6.5. Pointwise convergence of the real Fourier series: Dirichlet’s theorem 212 5.6.6. The Gibbs phenomenon and Cesàro summation 214 5.6.7. Speed of convergence to 0 of Fourier coefficients 214 5.6.8. Fourier transform in L 2 (T) and shift 218 5.7. Summary 219 Chapter 6. Bounded Linear Operators in Hilbert Spaces 221 6.1. Fundamental properties of bounded linear operators between normed vector spaces 223 6.1.1. Continuity of linear operators defined on a finite-dimensional normed vector space 226 6.2. The operator norm, convergence of operator sequences and Banach algebras 227 6.2.1. A classical example of a non-bounded linear operator on a vector space of infinite dimension 238 6.3. Invertibility of linear operators 239 6.4. The dual of a Hilbert space and the Riesz representation theorem 244 6.4.1. The scalar product induced on the dual of a Hilbert space 249 6.5. Bilinear forms, sesquilinear forms and associated quadratic forms 249 6.5.1. The Lax-Milgram theorem and its consequences 257 6.6. The adjoint operator: presentation and properties 261 6.7. Orthogonal projection operators in a Hilbert space 269 6.7.1. Bounded multiplication operators and their relation to orthogonal projectors 278 6.7.2. Geometric realization of orthogonal projection operators via orthonormal systems 280 6.8. Isometric and unitary operators 286 6.8.1. Characterizations of isometric and unitary operators 288 6.8.2. Relationship between isometric and unitary operators and orthonormal systems 293 6.9. The Fourier transform on S (Rn ), L 1 (Rn ) and L 2 (Rn ) 296 6.9.1. The invariance of the Schwartz space with r
• Edition: 1st
• Publisher Details: London : Wiley-ISTE
• Publication Date: 2021
• Extent: 1 online resource
• Subjects: 515.7; Functional analysis
• Languages: English
• ISBNs: 9781119851301
• Related ISBNs: 9781786306821
• Notes: Note: Description based on CIP data; resource not viewed.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.