Functional analysis for physics and engineering: an introduction
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CRC Press, Taylor & Francis Group
2020
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| Edition: | First issued in paperback |
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| Physical Description: | xv, 269 Seiten Illustrationen, Diagramme 24 cm |
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| 245 | 1 | 0 | |a Functional analysis for physics and engineering |b an introduction |c Hiroyuki Shima, University of Yamanashi, Japan |
| 250 | |a First issued in paperback | ||
| 264 | 1 | |a Boca Raton ; London ; New York |b CRC Press, Taylor & Francis Group |c 2020 | |
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| adam_text | Contents Dedication........................................................................................................................ v Preface............................................................................................................................ vii Chapter 1 Prologue.............................................................................................. 1 1.1 What Functional Analysis tells us........................................... 1 1.1.1 “Function” and “Functional analysis”.........................1 1.1.2 Infinite-dimensional spaces.......................................... 2 1.1.3 Relevance to quantum physics.....................................3 1.2 From perspective of the limit.....................................................4 1.2.1 Limit of a function........................................................4 1.2.2 What do you want to know: the limit’s value or existence?...................................................................... 6 1.2.3 Indirect approach to the limit........................................7 1.2.4 Function as the limit of mathematical operations...... 9 1.3 From perspective of infinite dimension.................................. 11 1.3.1 Topology of a space.................................................... 11 1.3.2 Length of a vector....................................................... 12 1.3.3 Size of a function......................................................... 14 1.3.4 Small infinity and large infinity.................................15 1.4 From perspective of quantum mechanical
theory...................16 1.4.1 Physical realization of the operator theory............... 16 1.4.2 Difficulty in choosing a basis in infinite dimension. 18 1.4.3 Plane waves are anomalous wave functions............. 19 Chapter 2 Topology............................................................................................. 21 2.1 Fundamentals............................................................................ 21 2.1.1 What is topology?........................................................21
x Functional Analysis for Physics and Engineering: An Introduction 2.1.2 What is closeness?......................................................22 2.1.3 Convergence of point sequences................................23 2.1.4 Open set and closed set...............................................26 2.1.5 Accumulation points....................................................29 2.2 Continuous mapping................................................................32 2.2.1 Relation between continuityand open sets................32 2.2.2 Relation between continuityand closed sets............ 35 2.2.3 Closure of a set............................................................ 38 2.3 Homeomorphism...................................................................... 41 2.3.1 Homeomorphic mapping........................................... 41 2.3.2 Revisited: What is topology?..................................... 43 2.3.3 Topological space theory........................................... 45 Chapter 3 Vector Space....................................................................................... 47 3.1 What is vector space?............................................................. 47 3.1.1 Beyond arrow-based representation.......................... 47 3.1.2 Axiom of vector spaces...............................................48 3.1.3 Example of vector spaces........................................... 50 3.2 Property of vector space.......................................................... 50 3.2.1 Inner
product.................................................................50 3.2.2 Geometry of inner product spaces............................ 54 3.2.3 Orthogonality and orthonormality.............................57 3.2.4 Linear independence...................................................59 3.2.5 Complete vector spaces...............................................61 3.3 Hierarchy of vector space........................................................ 62 3.3.1 From a mere collection of elements.......................... 62 3.3.2 Metric vector spaces...................................................63 3.3.3 Normed spaces............................................................ 64 3.3.4 Subspaces of normed spaces...................................... 65 3.3.5 Hilbert spaces...............................................................68 Chapter 4 Hilbert Space.....................................................................................71 4.1 Basis and completeness........................................................... 71 4.1.1 Basis of infinite-dimensiona! spaces......................... 71 4.1.2 Completeness of t2 spaces.......................................... 74 4.1.3 Completeness of I? spaces........................................ 75 4.1.4 Mean convergence in L2 spaces.................................75 4.2 Equivalence of L2 spaces with І2 spaces................................76 4.2.1 Generalized Fourier coefficient................................... 76 4.2.2 Riesz-Fisher theorem..................................................78 4.2.3 Isomorphism
between l2 and L2................................79
Contents Chapter 5 xi Tensor Space.......................................................................................81 5.1 Two faces of one tensor............................................................81 5.1.1 Tensor in practical science......................................... 81 5.1.2 Tensor in abstract mathematics..................................82 5.2 “Vector” as a linear function....................................................83 5.2.1 Linear function spaces................................................ 83 5.2.2 Dual spaces................................................................. 85 5.2.3 Equivalence between vectors and linear functions.. 86 5.3 Tensor as a multilinear function............................................. 87 5.3.1 Direct product of vector spaces..................................87 5.3.2 Multilinear functions...................................................88 5.3.3 Tensor product............................................................ 89 5.3.4 General definition of tensors...................................... 90 5.4 Component of tensor................................................................. 91 5.4.1 Basis of a tensor space............................................... 91 5.4.2 Transformation law of tensors...................................92 5.4.3 Natural isomorphism...................................................93 Chapter 6 Lebesgue Integral.............................................................................. 97 6.1 Motivation and
merits...............................................................97 6.1.1 Merits of studying Lebesgue integral....................... 97 6.1.2 Closer look to Riemann integral................................98 6.2 Measure theory....................................................................... 100 6.2.1 Measure...................................................................... 100 6.2.2 Lebesgue measure..................................................... 100 6.2.3 Countable set..............................................................102 6.3 Lebesgue integral....................................................................105 6.3.1 What is Lebesgue integral?....................................... 105 6.3.2 Riemann vs. Lebesgue..............................................106 6.3.3 Property of Lebesgue integral.................................. 107 6.3.4 Concept of “almost everywhere”.............................108 6.4 Lebesgue convergence theorem.............................................110 6.4.1 Monotone convergence theorem..............................110 6.4.2 Dominated convergence theorem............................. 111 6.4.3 Remarks on the dominated convergence theorem .113 6.5 LP space................................................................................... 115 6.5.1 Essence of IP space................................................... 115 6.5.2 Holder’s inequality...................... ............................ 116 6.5.3 Minkowski’s inequality.............................................117 6.5.4 Completeness of IP
space........................................ 118
xii Chapter 7 Functional Analysis for Physics and Engineering: An Introduction Wavelet............................................................................................. 121 7.1 Continuous wavelet analysis...................................................121 7.1.1 What is wavelet?........................................................121 7.1.2 Wavelet transform..................................................... 123 7.1.3 Correlation between wavelet and signal.................125 7.1.4 Contour plot of wavelet transform...........................125 7.1.5 Inverse wavelet transform.........................................127 7.1.6 Noise reduction..........................................................128 7.2 Discrete wavelet analysis........................................................130 7.2.1 Discrete wavelet transform...................................... 130 7.2.2 Complete orthonormal wavelet.................................131 7.3 Wavelet space.......................................................................... 133 7.3.1 Multiresolution analysis.............................................133 7.3.2 Orthogonal decomposition........................................135 7.3.3 Orthonormal basis construction...............................136 7.3.4 Two-scale relation..................................................... 137 7.3.5 Mother wavelet..........................................................139 7.3.6 Multiresolution representation................................. 139 Chapter 8
Distribution....................................................................................... 141 8.1 Motivation and merits............................................................. 141 8.1.1 Overcoming the confusing concept: “sham function”.................................................................... 141 8.1.2 Merits of introducing “Distribution”...................... 143 8.2 Establishing the concept of distribution............................... 143 8.2.1 Inconsistency hidden in the б-function................... 143 8.2.2 How to resolve the inconsistency.............................144 8.2.3 Definition of distribution...........................................146 8.3 Examples of distribution........................................................ 146 8.3.1 Dirac’s delta function................................................ 146 8.3.2 Heaviside’s step function..........................................147 8.3.3 Cauchy’s principal value as a distribution..............147 8.3.4 Remarks on the distribution PV±............................149 8.3.5 Properties of distribution...........................................149 8.3.6 Utility of distribution in solving differential equations.................................................................... 150 8.4 Theory of distribution.............................................................151 8.4.1 Rapidly decreasing function.................................... 151 8.4.2 Space of rapidly decreasing functions S(1R)...........152 8.4.3 Tempered distribution...............................................155 8.4.4
Distribution defined by integrable function............155 8.4.5 Identity between function and distribution.............158 8.5 Mathematical manipulation of distribution..........................159 8.5.1 Distributional derivative............................................. 159
xiii Contents 8.5.2 8.53 Chapter 9 Completion........................................................................................169 9.1 9.2 9.3 Chapter 10 Fourier transformation using distribution.............. 162 Weak solution to differential equation................... 166 Completion of number space................................................169 9.1.1 To make it complete artificially............................... 169 9.1.2 Incomplete space Q..................................................170 9.1.3 Creating ]R from Q.................................................... 174 Completion of function space................................................175 9.2.1 Continuous function spaces are incomplete...........175 9.2.2 Distance between continuous functions.................. 176 9.2.3 Completion from C to l)......................................... 177 9.2.4 Strategy for completion.............................................178 Sobolev space......................................................................... 179 9.3.1 What is Sobolev space?............................................179 9.3.2 Remarks on generation of Sobolev space............... 180 9.3.3 Application to Laplace operator manipulation.......181 9.3.4 Application in solving differential equations..........182 Operator..............................................................................................185 10.1 Classification of operators..................................................... 185 10.1.1 Basic premise........................................................... 185 10.1.2
Norms and completeness.........................................186 10.1.3 Continuous operators................................................187 10.1.4 Completely continuous operators...........................188 10.1.5 Non-continuous operators........................................ 188 10.1.6 Closed operators....................................................... 189 10.2 Essence of operator theory.................................................... 190 10.2.1 Definition of linear operator.................................... 190 10.2.2 Operator’s domain..................................................... 191 10.2.3 Definition of continuous linear operator................. 192 10.2.4 Definition of bounded operator................................195 10.2.5 Equivalence between continuity and boundedness 196 10.2.6 Artificial downscale of operator domain................. 198 10.3 Preparation toward eigenvalue-like problem.......................199 10.3.1 Spectrum................................................................... 199 10.3.2 Self-adjoint operator................................................ 200 10.3.3 Definition of self-adjoint operator.......................... 201 10.3.4 Compact or non-compact?....................................... 203 10.3.5 Loss of eigenvectors in infinite dimension..............204 10.4 Eigenvalue of “completely continuous” operator................206 10.4.1 Condition imposed on continuous operator.......... 206 10.4.2 Definition of completely continuous operator....... 208 10.4.3 Decomposition of Hilbert
space..............................209
Functional Analysis for Physics and Engineering: An Introduction xjv 10.4.4 Eigenvalue of completely continuous operator..... 210 10.5 Spectrum of “continuous” operator..................................... 212 10.5.1 Consequence of loss of complete continuity........ 212 10.5.2 Approximate eigenvector of continuous operator..213 10.5.3 Set of continually changing projection operators ..214 10.5.4 Spectral decomposition of operator.......................217 10.6 Practical importance of non-continuous operators.............218 10.6.1 “Non-commuting” means “non-continuous”........218 10.6.2 Ubiquity of non-commuting operators................... 220 Appendix A Real Number Sequence..................................................................223 A. 1 Convergence of real sequence.............................................. 223 A.2 Bounded sequence................................................................ 224 A.3 Uniqueness of the limit of real sequence.............................225 Appendix В Cauchy Sequence.............................................................................. 227 B. 1 What is Cauchy sequence?................................................... 227 В .2 Cauchy criterion for real number sequence.........................228 Appendix C Real Number Series.......................................................................... 231 C. 1 Limit of real number series................................................... 231 C.2 Cauchy criterion for real number series............................... 232 Appendix D Continuity and Smoothness of
Function.......................................235 D. 1 D.2 D.3 D.4 Appendix E Limit of function.................................................................... 235 Continuity of function............................................................236 Derivative of fünction............................................................238 Smooth function..................................................................... 240 Function Sequence.............................................................................241 E. 1 Pointwise convergence........................................................... 241 E.2 Uniform convergence.............................................................243 E.3 Cauchy criterion for function series..................................... 245 Appendix F Uniformly Convergent Sequence of Functions............................ 247 F. 1 F.2 F.3 Appendix G Continuity of the limit function............................................247 Integrability of the limit function......................................... 249 Differentiability of the limit function...................................249 Function Series................................................................................253 G. 1 Infinite series of functions..................................................... 253 G.2 Properties of uniformly convergent series of functions..... 254
Contents Appendix H XV Matrix Eigenvalue Problem.............................................................257 H. 1 H.2 Appendix I Eigenvalue and eigenvector................................................. 257 Hermite matrix.......................................................................258 Eigenspace Decomposition............................................................. 261 I.1 1.2 Eigenspace of matrix............................................................ 261 Direct sum decomposition....................................................263 Index............................................................................................................................. 265
This book provides an introduction to functional analysis for non-experts in mathematics. As such, it is distinct from most other books on the subject that are intended for mathematicians. Concepts are explained concisely with visual materials, making it accessible for those unfamiliar with include: graduate-level topology, vector mathematics. spaces, tensor Topics spaces, Lebesgue integrals, and operators, to name a few. Two central issues — the theory of Hilbert space and the operator theory — and how they relate to quantum physics are covered extensively. Each chapter explains, concisely, the purpose of the specific topic and the benefit of understanding it. Researchers and graduate students in information physics, mechanical science will engineering, and benefit from this view of functional analysis. 9780367737382 CRC Press Taylor Francis Group an informa business www.crcpress.com ДНЯ Խ
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| adam_txt |
Contents Dedication. v Preface. vii Chapter 1 Prologue. 1 1.1 What Functional Analysis tells us. 1 1.1.1 “Function” and “Functional analysis”.1 1.1.2 Infinite-dimensional spaces. 2 1.1.3 Relevance to quantum physics.3 1.2 From perspective of the limit.4 1.2.1 Limit of a function.4 1.2.2 What do you want to know: the limit’s value or existence?. 6 1.2.3 Indirect approach to the limit.7 1.2.4 Function as the limit of mathematical operations. 9 1.3 From perspective of infinite dimension. 11 1.3.1 Topology of a space. 11 1.3.2 Length of a vector. 12 1.3.3 Size of a function. 14 1.3.4 Small infinity and large infinity.15 1.4 From perspective of quantum mechanical
theory.16 1.4.1 Physical realization of the operator theory. 16 1.4.2 Difficulty in choosing a basis in infinite dimension. 18 1.4.3 Plane waves are anomalous wave functions. 19 Chapter 2 Topology. 21 2.1 Fundamentals. 21 2.1.1 What is topology?.21
x Functional Analysis for Physics and Engineering: An Introduction 2.1.2 What is closeness?.22 2.1.3 Convergence of point sequences.23 2.1.4 Open set and closed set.26 2.1.5 Accumulation points.29 2.2 Continuous mapping.32 2.2.1 Relation between continuityand open sets.32 2.2.2 Relation between continuityand closed sets. 35 2.2.3 Closure of a set. 38 2.3 Homeomorphism. 41 2.3.1 Homeomorphic mapping. 41 2.3.2 Revisited: What is topology?. 43 2.3.3 Topological space theory. 45 Chapter 3 Vector Space. 47 3.1 What is vector space?. 47 3.1.1 Beyond arrow-based representation. 47 3.1.2 Axiom of vector spaces.48 3.1.3 Example of vector spaces. 50 3.2 Property of vector space. 50 3.2.1 Inner
product.50 3.2.2 Geometry of inner product spaces. 54 3.2.3 Orthogonality and orthonormality.57 3.2.4 Linear independence.59 3.2.5 Complete vector spaces.61 3.3 Hierarchy of vector space. 62 3.3.1 From a mere collection of elements. 62 3.3.2 Metric vector spaces.63 3.3.3 Normed spaces. 64 3.3.4 Subspaces of normed spaces. 65 3.3.5 Hilbert spaces.68 Chapter 4 Hilbert Space.71 4.1 Basis and completeness. 71 4.1.1 Basis of infinite-dimensiona! spaces. 71 4.1.2 Completeness of t2 spaces. 74 4.1.3 Completeness of I? spaces. 75 4.1.4 Mean convergence in L2 spaces.75 4.2 Equivalence of L2 spaces with І2 spaces.76 4.2.1 Generalized Fourier coefficient. 76 4.2.2 Riesz-Fisher theorem.78 4.2.3 Isomorphism
between l2 and L2.79
Contents Chapter 5 xi Tensor Space.81 5.1 Two faces of one tensor.81 5.1.1 Tensor in practical science. 81 5.1.2 Tensor in abstract mathematics.82 5.2 “Vector” as a linear function.83 5.2.1 Linear function spaces. 83 5.2.2 Dual spaces. 85 5.2.3 Equivalence between vectors and linear functions. 86 5.3 Tensor as a multilinear function. 87 5.3.1 Direct product of vector spaces.87 5.3.2 Multilinear functions.88 5.3.3 Tensor product. 89 5.3.4 General definition of tensors. 90 5.4 Component of tensor. 91 5.4.1 Basis of a tensor space. 91 5.4.2 Transformation law of tensors.92 5.4.3 Natural isomorphism.93 Chapter 6 Lebesgue Integral. 97 6.1 Motivation and
merits.97 6.1.1 Merits of studying Lebesgue integral. 97 6.1.2 Closer look to Riemann integral.98 6.2 Measure theory. 100 6.2.1 Measure. 100 6.2.2 Lebesgue measure. 100 6.2.3 Countable set.102 6.3 Lebesgue integral.105 6.3.1 What is Lebesgue integral?. 105 6.3.2 Riemann vs. Lebesgue.106 6.3.3 Property of Lebesgue integral. 107 6.3.4 Concept of “almost everywhere”.108 6.4 Lebesgue convergence theorem.110 6.4.1 Monotone convergence theorem.110 6.4.2 Dominated convergence theorem. 111 6.4.3 Remarks on the dominated convergence theorem .113 6.5 LP space. 115 6.5.1 Essence of IP space. 115 6.5.2 Holder’s inequality. . 116 6.5.3 Minkowski’s inequality.117 6.5.4 Completeness of IP
space. 118
xii Chapter 7 Functional Analysis for Physics and Engineering: An Introduction Wavelet. 121 7.1 Continuous wavelet analysis.121 7.1.1 What is wavelet?.121 7.1.2 Wavelet transform. 123 7.1.3 Correlation between wavelet and signal.125 7.1.4 Contour plot of wavelet transform.125 7.1.5 Inverse wavelet transform.127 7.1.6 Noise reduction.128 7.2 Discrete wavelet analysis.130 7.2.1 Discrete wavelet transform. 130 7.2.2 Complete orthonormal wavelet.131 7.3 Wavelet space. 133 7.3.1 Multiresolution analysis.133 7.3.2 Orthogonal decomposition.135 7.3.3 Orthonormal basis construction.136 7.3.4 Two-scale relation. 137 7.3.5 Mother wavelet.139 7.3.6 Multiresolution representation. 139 Chapter 8
Distribution. 141 8.1 Motivation and merits. 141 8.1.1 Overcoming the confusing concept: “sham function”. 141 8.1.2 Merits of introducing “Distribution”. 143 8.2 Establishing the concept of distribution. 143 8.2.1 Inconsistency hidden in the б-function. 143 8.2.2 How to resolve the inconsistency.144 8.2.3 Definition of distribution.146 8.3 Examples of distribution. 146 8.3.1 Dirac’s delta function. 146 8.3.2 Heaviside’s step function.147 8.3.3 Cauchy’s principal value as a distribution.147 8.3.4 Remarks on the distribution PV±.149 8.3.5 Properties of distribution.149 8.3.6 Utility of distribution in solving differential equations. 150 8.4 Theory of distribution.151 8.4.1 Rapidly decreasing function. 151 8.4.2 Space of rapidly decreasing functions S(1R).152 8.4.3 Tempered distribution.155 8.4.4
Distribution defined by integrable function.155 8.4.5 Identity between function and distribution.158 8.5 Mathematical manipulation of distribution.159 8.5.1 Distributional derivative. 159
xiii Contents 8.5.2 8.53 Chapter 9 Completion.169 9.1 9.2 9.3 Chapter 10 Fourier transformation using distribution. 162 Weak solution to differential equation. 166 Completion of number space.169 9.1.1 To make it complete artificially. 169 9.1.2 Incomplete space Q.170 9.1.3 Creating ]R from Q. 174 Completion of function space.175 9.2.1 Continuous function spaces are incomplete.175 9.2.2 Distance between continuous functions. 176 9.2.3 Completion from C to l). 177 9.2.4 Strategy for completion.178 Sobolev space. 179 9.3.1 What is Sobolev space?.179 9.3.2 Remarks on generation of Sobolev space. 180 9.3.3 Application to Laplace operator manipulation.181 9.3.4 Application in solving differential equations.182 Operator.185 10.1 Classification of operators. 185 10.1.1 Basic premise. 185 10.1.2
Norms and completeness.186 10.1.3 Continuous operators.187 10.1.4 Completely continuous operators.188 10.1.5 Non-continuous operators. 188 10.1.6 Closed operators. 189 10.2 Essence of operator theory. 190 10.2.1 Definition of linear operator. 190 10.2.2 Operator’s domain. 191 10.2.3 Definition of continuous linear operator. 192 10.2.4 Definition of bounded operator.195 10.2.5 Equivalence between continuity and boundedness 196 10.2.6 Artificial downscale of operator domain. 198 10.3 Preparation toward eigenvalue-like problem.199 10.3.1 Spectrum. 199 10.3.2 Self-adjoint operator. 200 10.3.3 Definition of self-adjoint operator. 201 10.3.4 Compact or non-compact?. 203 10.3.5 Loss of eigenvectors in infinite dimension.204 10.4 Eigenvalue of “completely continuous” operator.206 10.4.1 Condition imposed on continuous operator. 206 10.4.2 Definition of completely continuous operator. 208 10.4.3 Decomposition of Hilbert
space.209
Functional Analysis for Physics and Engineering: An Introduction xjv 10.4.4 Eigenvalue of completely continuous operator. 210 10.5 Spectrum of “continuous” operator. 212 10.5.1 Consequence of loss of complete continuity. 212 10.5.2 Approximate eigenvector of continuous operator.213 10.5.3 Set of continually changing projection operators .214 10.5.4 Spectral decomposition of operator.217 10.6 Practical importance of non-continuous operators.218 10.6.1 “Non-commuting” means “non-continuous”.218 10.6.2 Ubiquity of non-commuting operators. 220 Appendix A Real Number Sequence.223 A. 1 Convergence of real sequence. 223 A.2 Bounded sequence. 224 A.3 Uniqueness of the limit of real sequence.225 Appendix В Cauchy Sequence. 227 B. 1 What is Cauchy sequence?. 227 В .2 Cauchy criterion for real number sequence.228 Appendix C Real Number Series. 231 C. 1 Limit of real number series. 231 C.2 Cauchy criterion for real number series. 232 Appendix D Continuity and Smoothness of
Function.235 D. 1 D.2 D.3 D.4 Appendix E Limit of function. 235 Continuity of function.236 Derivative of fünction.238 Smooth function. 240 Function Sequence.241 E. 1 Pointwise convergence. 241 E.2 Uniform convergence.243 E.3 Cauchy criterion for function series. 245 Appendix F Uniformly Convergent Sequence of Functions. 247 F. 1 F.2 F.3 Appendix G Continuity of the limit function.247 Integrability of the limit function. 249 Differentiability of the limit function.249 Function Series.253 G. 1 Infinite series of functions. 253 G.2 Properties of uniformly convergent series of functions. 254
Contents Appendix H XV Matrix Eigenvalue Problem.257 H. 1 H.2 Appendix I Eigenvalue and eigenvector. 257 Hermite matrix.258 Eigenspace Decomposition. 261 I.1 1.2 Eigenspace of matrix. 261 Direct sum decomposition.263 Index. 265
This book provides an introduction to functional analysis for non-experts in mathematics. As such, it is distinct from most other books on the subject that are intended for mathematicians. Concepts are explained concisely with visual materials, making it accessible for those unfamiliar with include: graduate-level topology, vector mathematics. spaces, tensor Topics spaces, Lebesgue integrals, and operators, to name a few. Two central issues — the theory of Hilbert space and the operator theory — and how they relate to quantum physics are covered extensively. Each chapter explains, concisely, the purpose of the specific topic and the benefit of understanding it. Researchers and graduate students in information physics, mechanical science will engineering, and benefit from this view of functional analysis. 9780367737382 CRC Press Taylor Francis Group an informa business www.crcpress.com ДНЯ Խ |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV047066858 |
| illustrated | Illustrated |
| index_date | 2024-07-03T16:12:51Z |
| indexdate | 2024-07-10T09:01:39Z |
| institution | BVB |
| isbn | 9780367737382 9781482223019 0367737388 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032473942 |
| oclc_num | 1238067803 |
| open_access_boolean | |
| owner | DE-706 DE-703 |
| owner_facet | DE-706 DE-703 |
| physical | xv, 269 Seiten Illustrationen, Diagramme 24 cm |
| publishDate | 2020 |
| publishDateSearch | 2020 |
| publishDateSort | 2020 |
| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| spelling | Shima, Hiroyuki Verfasser (DE-588)141263954 aut Functional analysis for physics and engineering an introduction Hiroyuki Shima, University of Yamanashi, Japan First issued in paperback Boca Raton ; London ; New York CRC Press, Taylor & Francis Group 2020 xv, 269 Seiten Illustrationen, Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier Physik (DE-588)4045956-1 gnd rswk-swf Ingenieurwissenschaften (DE-588)4137304-2 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Functional analysis Functional equations Mathematical physics Engineering mathematics (DE-588)4123623-3 Lehrbuch gnd-content Funktionalanalysis (DE-588)4018916-8 s Physik (DE-588)4045956-1 s Ingenieurwissenschaften (DE-588)4137304-2 s DE-604 Erscheint auch als Online-Ausgabe 978-0-429-06816-4 Erscheint auch als Online-Ausgabe 978-1-4822-2303-3 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032473942&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032473942&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Shima, Hiroyuki Functional analysis for physics and engineering an introduction Physik (DE-588)4045956-1 gnd Ingenieurwissenschaften (DE-588)4137304-2 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| subject_GND | (DE-588)4045956-1 (DE-588)4137304-2 (DE-588)4018916-8 (DE-588)4123623-3 |
| title | Functional analysis for physics and engineering an introduction |
| title_auth | Functional analysis for physics and engineering an introduction |
| title_exact_search | Functional analysis for physics and engineering an introduction |
| title_exact_search_txtP | Functional analysis for physics and engineering an introduction |
| title_full | Functional analysis for physics and engineering an introduction Hiroyuki Shima, University of Yamanashi, Japan |
| title_fullStr | Functional analysis for physics and engineering an introduction Hiroyuki Shima, University of Yamanashi, Japan |
| title_full_unstemmed | Functional analysis for physics and engineering an introduction Hiroyuki Shima, University of Yamanashi, Japan |
| title_short | Functional analysis for physics and engineering |
| title_sort | functional analysis for physics and engineering an introduction |
| title_sub | an introduction |
| topic | Physik (DE-588)4045956-1 gnd Ingenieurwissenschaften (DE-588)4137304-2 gnd Funktionalanalysis (DE-588)4018916-8 gnd |
| topic_facet | Physik Ingenieurwissenschaften Funktionalanalysis Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032473942&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032473942&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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