Zeidler E. — Applied Functional Analysis: Applications to Mathematical Physics :: Электронная библиотека попечительского совета мехмата МГУ

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Zeidler E. — Applied Functional Analysis: Applications to Mathematical Physics

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Название: Applied Functional Analysis: Applications to Mathematical Physics

Автор: Zeidler E.

Аннотация:

This is a incredible book on applied functional analyses. Every topic is motivated with an applied problem. The definitions are motivated either by the aplication or by the subsequent use. There are remainders showing you the inteconections between the subjects and finally the index and the Symbols index are both complete and very usefull. The book is not complete. However he missing subjects usually are in the other colection by the same author.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1995

Количество страниц: 509

Добавлена в каталог: 04.02.2014

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Предметный указатель

$(u|v)_{1, 2}$ , inner product on the Sobolev space $W_{2}^{1}(G)$ , $(u|v)_{1, 2}:=\int_{G}(uv+\sum\limits_{j=1}^{N}\partial_{j}u\partial_{j}v)dx$       120
$(u|v)_{2}$ , inner product on the Lebesgue spaces $L_{2}(G)$ and $L_{2}^{\mathbb{C}}(G)$ , $(u|v)_{2}:=\int_{G}\overline{u(x)}v(x)dx$       114
$(\cdot|\cdot)_{E}$ , energetic inner product      273
-automorphism      358
-Homomorphism      358
-isomorphism      358
$A:S\subseteq X \rightarrow Y$ , operator from the set S into the set Y, where $S \subseteq Y$       17
$A\subseteq B$ , the operator B is an extension of the operator A      260
$A^{-1}$ , inverse operator to A      17
$A^{-1}(T)$ , preimage of the set T, $A^{-1}(T):=\{x:Ax\in T\}$       17
$C^{k}[a, b]$ , $C^{k}(G)$ , $C^{k}(\bar{G})$ , $C^{\infty}(G)$ , $C^{k}(G)_{\mathbb{C}}$ $(C^{0}(\bar{G}):=C(\bar{G}))$ , spaces of smooth functions      96 116
$C^{\alpha}[a, b]$ , $C^{k, \alpha}[a, b]$ , $C^{\alpha}(\bar{G})$ , $C^{k, \alpha}(\bar{G})$ $(C^{\alpha}(\bar{G}) = C^{0, \alpha}(\bar{G}))$ , spaces of Hoelder continuous functions      95ff
$C_{0}^{\infty}(G)$ (or $\mathcal{D}(G)$ ), $\mathcal{S}$ , spaces of smooth functions      116 214
$L^{\bot}$ , orthogonal complement to the linear subspace L      165
$L_{2}(a, b)$ , $L_{2}(G)$ , $L_{2}^{\mathbb{K}}(G)$ $(L_{2}(G):=L_{2}^{\mathbb{K}}(G) if \mathbb{K} = \mathbb{R})$ , spaces of integrable functions (Lebesgue spaces)      130 114
$M\oplus L$ , orthogonal direct sum ( $M\oplus L$ , where $L=M^{\bot}$ )      165
$u^{'}(t)$ , derivative of an operator function u = u(t) at time t      80
$U_{\varepsilon}(p)$ , $\varepsilon$ -neighborhood of the point p in a normed space, $U_{\varepsilon}(p):=\{x\in X: \|x-p\|<\varepsilon\}$       15
$W_{2}^{1}(G)$ , $\r{W}_{2}^{1}(G)$ , Sobolev spaces      131 132
$X\otimes Y$ , tensor product      224
$X_{E}$ , energetic space      273
$X_{\mathbb{C}}$ , complexification of the linear space X      98
$\alpha S$ , the product $\alpha S:=\{\alpha x:x\in S\}$ , $\alpha\in\mathbb{R},\mathbb{C}$       7
$\bar{A}$ , closure of the linear operator A      415
$\bar{co}$ S, closed convex hull of the set S      47
$\bar{S}$ , closure of the set S      31
$\Delta f$ , Laplacian, $\Delta f:=\sum\limits_{n=1}^{N}\partial_{j}^{2}f$       125
$\delta$ , the delta distribution      161
$\delta(x)$ , the Dirac delta function      158
$\frac{\partial}{\partial n}$ , derivative in the direction of the exterior normal      181
$\lim\limits_{n\rightarrow\infty}x_{n} = x$ (or $x_{n}\rightarrow x$ as $n\rightarrow\infty$ ), the sequence $(x_{n})$ converges to the point x      9
$\mathbb{K}^{\infty}$ , $l_{\infty}^{\mathbb{K}}$ , $l_{2}^{\mathbb{K}}$ $(l_{2}:=l_{2}^{\mathbb{K}} if \mathbb{K} = \mathbb{R})$ , spaces of sequences      95 177
$\mathcal{D}^{'}(G)$ , $\mathcal{S}^{'}$ , spaces of distributions      160 219
$\partial S$ , boundary of the set S      31
$\partial^{\alpha}f$ , $\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}\cdots\partial_{N}^{\alpha_{N}}f$ , where $\alpha = (\alpha_{1},\ldots,\alpha_{N})$ (the classical symbols are also used for the derivatives of generalized functions)      159
$\rho(A)$ , resolvent set of the linear operator A      83
$\sigma(A)$ , spectrum of the linear operator A      83
$\sum\limits_{n=1}^{\infty}u_{n}$ , infinite series in a Banach space      76
$\|A\|$ , norm of the linear operator A      70
$\|f\|$ , norm of the functional f      75
$\|u\|_{2}$ , norm on the Lebesgue spaces $L_{2}(G)$ and $L_{2}^{\mathbb{C}}(G)$ , $\|u\|_{2}:=(u|u)^{\frac{1}{2}}=(\int_{G}|u(x)|^{2}dx)^{\frac{1}{2}}$       114
$\|x\|$ , norm of x      7
$\|\cdot\|_{E}$ , energetic norm      273
$|x|_{\infty}$ , special norm, $|x|_{\infty}:=\sup \limits_{n} |x_{n}|$       11
$|\alpha|$ , the sum $\alpha_{1}+\cdots+\alpha_{N}$       159
(x|y), inner product      105
<x|y>, Euclidean inner product, $<x|y>:=\sum\limits_{n=1}^{N}\overline{x}_{n}y_{n}$ ( $\overline{y}_{j}$ conjugate complex number to $y_{j}$ )      109
A posteriori error estimate      19
A priori error estimate      19
A priori estimates      64
A*, adjoint operator to the linear operator A      263
AB (or $A\circ B$ ), the product of the operators A and B, (AB)(u):=A(Bu)      28
Absolute continuity      437
Absolute integrability      435
Absolute temperature      352
Absolutely convergent      76
Abstract boundary-eigenvalue problem      255
Abstract boundary-value problem      254
Abstract Dirichlet problem      282
Abstract Fourier series      200
Abstract heat equation      255 306
Abstract Schroedinger equation      256 323
Abstract setting for quantum mechanics      327
Abstract setting of quantum statistics      348
Abstract wave equation      256 310
action      393
Addition theorems      305
Adjoint operator      263
Admissible paths      392
Admissible sequence      274
Algebraic approach to quantum statistics      357
Almost all      431
Almost everywhere      431
Annihilation operator      366
Anticommutation relations      367
Antilinear      169
Appolonius' identity      178
Arzela — Ascoli theorem      35
Asymptotically free      368
Balls      16 94
Banach algebra      76
Banach fixed-point theorem      18
Banach space      10
Barycenter      47
Barycentric subdivision      49
Basic equation of quantum statistics      351
Basis      42
Beauty of functional analysis      256
Bernstein polynomials      86
Bessel inequality      202
big bang      357
Bijective      17
Bilinear form      120
Black holes      340
Bolzano — Weierstrass theorem      34
Born approximation      403 404
Bose — Einstein statistics      355
Bosonic Fock space      364
Bosons      363
Bound states      372 394 396
Boundary      31
Boundary point      31
Boundary-eigenvalue problem      245 285 316
Boundary-value problem      125
Bounded      33
Bounded bilinear form      120
Bounded orbits      421
Bounded sequence      9
Brouwer fixed-point theorem      53
Brownian motion      381
C*-algebra      357
Calculus of variations      117 126
Cauchy sequence      10
Cayley transform      423
Characteristic equation      94
Charge density      180
chemical potential      352
Chronological operator      395
Classical function spaces      95 97
Classical Schwarz inequality      12
Closed      15
Closed balls      16
Closed convex hull      31
Closed linear subspace      30
Closure      31
Closure of an operator      414
co S, convex hull of the set S      31
Commutation relations      365
Commutative C*-algebra      358
Compact      90
Compact embedding      96 261
Compact operator      39
Compact set      33
Compactness      33
complete      10
Complete orthonormal system      200 209 210 223 232 247 374
Complete system of generalized eigenfunctions      346
Completeness relation      374
Completeness theorem      222
Complex linear space      4
Complex normed space      7
Complexification      98
Complexification of real Hilbert spaces      178
Composite states of elementary particles      227
Conservation of energy      336 404
Continuity      26
continuous      27
Continuous Dirac calculus      375
Continuous embedding      261
Continuous spectrum      409

Contraction principle      19
Convergence      9
Convex      29
Convex hull      31
Convexity      29
Convolution      182
Coulomb force      180
Coulomb potential      420
Countable      84
Creation operators      364
Critical point      321
C[a, b], $C(\bar{G})$ , spaces of continuous functions      14 116
D(A) (or dom A), domain of definition of the operator A      17
Defect indices      423
Deflection of a string      157
Degenerate kernels      251
Degrees of freedom      229
Dense      84
density      84 189 222
Derivative      80
Diagonal sequence      36
diam S, diameter of the set S      47
Diameter      47
Dielectricity constant      420
Diffeomorphism      435
Differential operator      267
Diffusion      379 386
Diffusion equation      379
dim X, dimension of the linear space X      6
DIMENSION      6
Dirac $\delta$ -distribution      161
Dirac $\delta$ -function      158 346
Dirac calculus      222 373
Dirac function      375
Dirichlet principle      101 123 138
Dirichlet problem      125
Discrete Dirac calculus      374
Discrete spectrum      409
Disk      28
dispersion      328
Dispersion of the energy      352
Dispersion of the particle number      352
Distance      7 47
Distributions      160
Domain of definition      17
Dominated convergence      437
Dual maximum problem      169
Dual space      75
Duality map      169 279
Duality of quadratic variational problems      169
Duality theory      142
Dynamical systems      303
Dynamics      359
Dynamics of quantum systems      328
Dynamics of statistical systems      349
Dynamics of the harmonic oscillator      341
Dyson formula      395
Eigenfunction      247 340
Eigenoscillations      198
Eigenoscillations of the string      317
Eigensolution      230 241
Eigenspace      230
Eigenstate      329
Eigenvalue      83 230
Eigenvalue problem      283
Eigenvector      230
Elastic energy      256
Elasticity      145
electric field      180
Electric field of a charged point      180
Electrostatic potential      180
Electrostatics      183
Embedding theorems      262
Energetic extension      279
Energetic inner product      123 144 273
Energetic norm      144 273
Energetic space      144 154 273
Energy      309
Energy conservation      303 310 313
Energy of the harmonic oscillator      309
entropy      349 353
Equicontinuous      35
Equivalent norms      42 99
Error estimate      19 142
Error estimates via duality      142
Essential spectrum      84 372
Euclidean norm      12 109
Euclidean strategy in quantum physics      379
Euler — Lagrange equation      125
Expansion of our universe      340 357
Exponential function      78
ext S, exterior of the set S      31
Extension      261
Extension Principle      213
Exterior      31
Exterior point      31
Fatou lemma      110 438
Fermi — Dirac statistics      356
Fermionic Fock space      366
fermions      363
Feynman diagrams      400
Feynman formula      393
Feynman path integral      381 385 393
Feynman relation for transition amplitudes      397
Feynman — Kac formula      391
Finite $\varepsilon$ -net      38
Finite elements      151
Finite multiplicity      230
Finite-dimensional Banach spaces      42
Finite-dimensional space      6
Fixed point      19
Fock space      363
Force density      156
Formal hamiltonian      339
Fourier coefficients      195 200
Fourier integral      198
Fourier method      315 316
Fourier series      195 203 241 247
Fourier transform      216 376 378 413
Fourier transform of tempered generalized functions      219
Fredholm alternative      237 245 249 284 287
Fredholm integral operator      95
Free Hamiltonian      371
Friedrichs extension      258 280
Friedrichs extension in complex Hilbert spaces      419
Friedrichs' mollification      186
Fubini's theorem      439
Function      17
Functional calculus      293 331
Functions of bounded variation      440
Functions of self-adjoint operators      293
Fundamental solution      179 182 183 186 413
Fundamental theorem of calculus      119
G(A), graph of the operator A, $G(A):=\{(x, Ax):x\in D(A)\}$       414
Gauss method of least squares      197
Gauss theorem      119
Gaussian functions      223
Gelfand — Kostyuchenko theorem      424
Gelfand — Levitan — Marchenko integral equation      410
General position      45
Generalized boundary values      135 138
Generalized derivative      129
Generalized diffusion equation      386
Generalized Dirichlet problem      138
Generalized eigenfunctions      343 372 377
Generalized eigenvectors      424
Generalized Fourier series      195
Generalized functions      156 160
Generalized functions in mathematical physics      179
Generalized initial-value problem      185

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