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Morrison T.M. — Functional Analysis: An Introduction to Banach Space Theory

Morrison T.M. — Functional Analysis: An Introduction to Banach Space Theory

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Название: Functional Analysis: An Introduction to Banach Space Theory

Автор: Morrison T.M.

Аннотация:

A powerful introduction to one of the most active areas of theoretical and applied mathematics

This distinctive introduction to one of the most far-reaching and beautiful areas of mathematics focuses on Banach spaces as the milieu in which most of the fundamental concepts are presented. While occasionally using the more general topological vector space and locally convex space setting, it emphasizes the development of the reader’s mathematical maturity and the ability to both understand and "do" mathematics. In so doing, Functional Analysis provides a strong springboard for further exploration on the wide range of topics the book presents, including:

* Weak topologies and applications
* Operators on Banach spaces
* Bases in Banach spaces
* Sequences, series, and geometry in Banach spaces

Stressing the general techniques underlying the proofs, Functional Analysis also features many exercises for immediate clarification of points under discussion. This thoughtful, well-organized synthesis of the work of those mathematicians who created the discipline of functional analysis as we know it today also provides a rich source of research topics and reference material.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2000

Количество страниц: 359

Добавлена в каталог: 26.10.2010

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Предметный указатель

Daniell integral, "Fatuou's Lemma"      14
Daniell integral, dominated convergence theorem      14
Daniell integral, functional      12
Daniell integral, measurability      15
Daniell integral, monotone convergence theorem      14
Daniell integral, vector lattice      12
Daniell, P.J.      2 11 see
Davis — Figel — Johnson — Pelczynski Factorization Theorem      214—215
Davis, W.J.      214 see
Day, M.M.      322 340
de Branges, L.      152
Dense subspaces      see "Subspaces" "The
Descent of an operator      193
Diestel, J.      54 89 180 270 310 321 328 330
Dieudonne, J.      118 130 174
Differentiability of norm in dual and reflexivity      330
Differentiability of norm, in dual and reflexivity and smoothness      323—325
Differentiability of norm, in dual and reflexivity and support mappings      322—323
Differentiability of norm, in dual and reflexivity, Frechet      328 329—330
Differentiability of norm, in dual and reflexivity, Gateaux      323 324—325
Differentiability of norm, in dual and reflexivity, uniformly Frechet      330 331—333
Direct sum (product) of spaces      211—213
Direct sum decomposition      160—161
Direct sum decomposition, topological      161
Dixmier, X.      276
Dominated Convergence Theorem      14 35 58
Dor, L.      259
Dual space(s)      39 63—64
Dual space(s) in weak topology      120
Dual space(s), algebraic      64
Dual space(s), basis in reflexive spaces      297
Dual space(s), boundedly complete basis forces space to be      295—296
Dual space(s), calculation of, $(\mathbb{C}^{n})^{*}$       43
Dual space(s), calculation of, $(\mathbb{R}^{n})^{*}$       43
Dual space(s), calculation of, $c^{*}$       48—49
Dual space(s), calculation of, $c^{*}_{0}$       44—45
Dual space(s), calculation of, $l^{*}_{1}$       45—46
Dual space(s), calculation of, $L^{*}_{1}(\mu)$       54
Dual space(s), calculation of, $l^{*}_{p}$       46—48
Dual space(s), calculation of, $L^{*}_{p}(\mu)$       54—56
Dual space(s), calculation of, $l^{*}_{\infty}$       50
Dual space(s), calculation of, $\mathcal{C}(\mathcal{K}^{*})$       59—60
Dual space(s), coincidence of      48—49
Dual space(s), separable exactly when $U_{$ \mathcal{X}}$ weakly metrizable      137—139
Dual space(s), separable implies bounded sequences in X have weak Cauchy subsequences      139
Dual space(s), separable implies underlying space is      71
Duality and dual systems      118—121
Dunford — Pettis operator      305 306 307 308
Dunford — Pettis operator, are operator ideal      307
Dunford — Pettis operator, characterization of when weakly compact operator is      306—307
Dunford — Pettis operator, examples      308
Dunford — Pettis operator, implies strictly singular      307—308
Dunford — Pettis property      305 306 309
Dunford — Pettis Property and strictly cosingular operators      309
Dunford — Pettis Property, examples      305—306
Dunford, N.      142 156 181 187 see "Dunford "Dunford
Dunford-Pettis theorem      176 181 306
Eberlein — Smulian theorem      124 129 130—136 137 142 144 170 216 300 307
Eberlein, W.F.      127 130 see
Eigenvalues and eigenvectors      198—201
Enflo, P.      219
Equivalent bases      238
Equivalent bases, basic sequences      240
Equivalent bases, characterization of      239—240
Essentially bounded function(s)      35
Exposed point(s)      326—327
Extreme point(s)      145 146—149
Extreme point(s) and best approximations      149—150
Extreme point(s) and smooth points      324
Extreme point(s) and supporting sets      146
Extreme point(s) in $U^{*}_{x}$       150
Extreme point(s) in $\mathcal{C}(\mathcal{K}^{*})$       148—149
Extreme point(s), characterization of      145
Extreme point(s), existence of for compact convex sets      146—148
Extreme point(s), retrieving norm from      151
Faires, B.      330
Fatou's lemma      14
Figel, T.      214 see
Finite-dimensional space(s) and compact operators      182
Finite-dimensional space(s), all algebraically isomorphic      9
Finite-dimensional space(s), all topologically isomorphic      9—10
Finite-dimensional space(s), always complemented      161
Finite-dimensional space(s), compact sets in      10
Finite-dimensional space(s), compact unit ball      133
Finite-dimensional space(s), completeness      10
Finite-dimensional space(s), Hamel basis for      8
Finite-dimensional space(s), subspaces of      10
Frechet differentiability      see "Differentiability"
Frechet space(s)      218
Frechet, M.      218 see "Frechet
Fredholm operator theory      187 197
Friedberg, S.H.      8
Fubini's theorem      186
Functional(s)      39
Functional(s), coefficient      220
Functional(s), determining nbhds of $\theta$ in locally convex spaces      113—114
Functional(s), haif-spaces determined by      121
Functional(s), linear dependence of      119—120
Functional(s), Minkowski      114
Functional(s), separating in locally convex spaces      115—116 120—121
Functional(s), separating in normed linear spaces      111—112
Gantmacher — Nachamura Theorem      173 174 177 182
Gantmacher, V.      173 see
Gateaux differentiability      see "Differentiability"
Gelfand, I.M.      87
Giles, J.R.      322
Goldstine's Theorem      127—128 138 170 171 173 295 298
Goldstine, H.H.      128 see
Grothendieck space(s)      175 177 179 184 309
Grothendieck space(s), equivalents,      177—179
Grothendieck space(s), pointwise limit of compact operators on      184
Grothendieck, A.      130 174 175 218 275 280 305 309 see
Grymblyum — Nikol'skii Theorem      230—232 234 237 238 243 249 253 265
Haar measure(s)      90—94
Haar system for $L_{p}([0,1])$       234—237
Haar system for $L_{p}([0,1])$ and biorthogonal functionals      236—237
Haar system for $L_{p}([0,1])$ , boundedly complete      301
Haar system for $L_{p}([0,1])$ , shrinking      290
Haar, A.      90 92 234 see 1])$"/>
Hahn — Banach theorem      34 51 63 64 65—68 71 81 82 83—84 86 87 88 89 94 105 106 111—112 115 129 132 138 139 150 151 153 157 162 163 166 167 169 203 221 231 244 251 293 312 339
Hahn — Banach Theorem in locally convex spaces      115—116
Hahn — Banach Theorem, complex      83—84
Hahn — Banach Theorem, extension form      65—68
Hahn — Banach Theorem, separation form      111—112
Hahn, H.      64 104 277 see
Halmos, P.R.      35
Hamel basis      28 64 162 164 203 217 220 221 222
Hamel basis, uncountable in complete, infinite dimensional spaces      221—222
Hausdorff maximality principle      65 68 147
Heine — Borel theorem      10 125
Helly's theorem      63 104—105 318
Helly, E.      104 see
Hermann, R.      257
Hilbert cube      189
Hilbert space(s)      60—62 89 162—163 184—185 187 327 339—340
Hilbert space(s) and uniform convexity      339—340
Hilbert space(s), complemented subspaces      162—163
Hilbert — Schmidt operators      186
Hilbert, D.      181—184 see
Hoelder's inequality      29 30 31 47 54 103 186
Hoelder, E.      see "Holder's Inequality"
Holub, J.R.      321
Homeomorphism      22
Hsieh, M.      334
Huff, R.E.      312
Idempotent      159
Insel, A.J.      8
integral      see "Daniell integral"
Internal point(s)      109
James' space $\mathbf{J}$       73—75 220 232 237 257 258 270 290 292 293 299 301 302 303

James' space $\mathbf{J}$ , basis for      237—238
James' space $\mathbf{J}$ , basis for, not boundedly complete      301
James' space $\mathbf{J}$ , basis for, shrinking      290
James' space $\mathbf{J}$ , is isometrically isomorphic to dual      301—303
James' space $\mathbf{J}$ , is of codimension one in bidual      75 292—294
James' space $\mathbf{J}$ , is somewhat reflexive      257—258
James' space $\mathbf{J}$ , nonreflexivity      299
James' theorem      297—299 300 330 334
James, R.C.      73 75 220 237 270 286 290 294 297 310 313 314 317 319 322 387 see "James'
Johnson, W.B.      214 219 see
Jordan decomposition      55 60
Kadec, M.I.      218 334 337 338
Kakutani, S.      127 see
Karlin, S.      219
Kato, T.      202
Kershner, R.      93
Klee, V.      313 317 327
Knopp — Lorentz Summability Theorem      63 101—102
Knopp, K.      63 see
Koethe, G.      24 334
Krein — Milman theorem      107 127 142 146—148 149 151 153
Krein — Milman — Rutman theorem      242
Krein — Smulian Theorem      142—143 143—144 168 169 170 177 178
Krein — Smulian Theorem, closed convex hull of closed convex sets is weakly compact      143—144
Krein — Smulian Theorem, when convex sets are weak*-closed in $\mathcal{X}^{*}$       142—143
Krein, M.G.      144 146 148 241 242 see "Krein
Kronecker delta function,      225
Lifting problem      32
Lindenstrauss, J.      161 162 176 216 219 252 see
Linear operator(s)      see "Operator(s)"
Linear projection(s)      see "Projection(s)" "Complemented
Linear topological space(s)      see "Topological vector space(s)"
Liouville's theorem      85
Liouville, J.      85
Lipschitz function(s)      52
Lipschitz measures      52 180 see \mathcal{X})$"/>"
Locally convex space(s)      113—114
Locally convex space(s) as a product of Banach spaces      116—117
Locally convex space(s), existence of nonzero functionals on      113 121
Locally convex space(s), quasi-complete      319
Locally convex space(s), weak dual of      121
Locally convex topology      119
Lohman, R.H.      247
Lorentz, G.G.      63 see
Lusternick      241
Matrix maps      95—103
Matrix maps, conservative      100
Matrix maps, map bounded sequences to bounded sequences      97—98
Matrix maps, map convergent sequences to convergent sequences      98—99
Matrix maps, map summable sequences to summable sequences      102
Matrix maps, regular (permanent)      100
Mazur Selection Principle      247—248
Mazur's theorem      122 132 136 139 144 178 229 231 280 286 320 330
Mazur, S.      95 122 140 143 173 247 see "Mazur's "Mazur's
McArthur, C.W.      218 280
McShane, E.J.      334
McWilliams, R.D.      128 129
Mean value theorem      29
Measure(s), Baire      57
Measure(s), counting      56
Measure(s), existence of invarient      89—94
Measure(s), Lizschitzean      52
Measure(s), metric outer      90 93
Milman, D.P.      144 146 148 242 257 333 see "Krein "Krein
Milman, V.D.      257
Minkowski (gauge) functional(s)      109 110 111 112 113 114 116 212
Minkowski (gauge) functional(s), when continuous      114—116 212
Minkowski's inequality      30 31 73 74 236
Minkowski, H.      109 see "Minkowski
Minusinski, J.      77
Modulus of convexity      331 337 338
Modulus of convexity of Hilbert space      340
Monomorphism      202
Monotone Convergence Theorem      35 55
Moore, T.O.      140
Munroe, J.R.      265
Nagumo, M.      85
Nakamura, M.      173 see
Natural injection j      72 127—128 133 138—139 170—171 286
Nearest point(s)      326—327
Nets      3 5
Nets, weak and weak* convergence of      156
Neumann series of operators      190—191
Neumann, G      190
Nordlander, G.      340
Norm      9 17—18
Norm achieving functionals and reflexivity      313 317—319
Norm achieving functionals and weak compactness      319—321
Norm, $\parallel \cdot \parallel_{\infty}$       20
Norm, comparable      80 81
Norm, defined by functionals      70
Norm, Frechet differentiable      328—330
Norm, Gateaux differentiable      323—325
Norm, norm topology      22
Norm, operator (uniform)      39
Norm, positive homogeneous      18
Norm, seminorm      24
Norm, subadditive      18
Norm, uniformly Fredchet differentiable      330—333
Norm, variational      50
Normed linear space(s)      18
Normed linear space(s), reflexive always complete      75
Norming set      129 247
Norming set, when have countable one in dual      129
Null space (kernel) of operator      160
Null space (kernel) of operator and complementation      195
Null space (kernel) of operator, properties of      194—198
Open-Mapping Theorem      63 78—80 81 82 103 104 105 117 167 208 289
Operator ideal(s)      171—172 183 184 193 205 207 208 307
Operator ideal(s) in reflexive $l_{p}$ -spaces      184
Operator ideal(s), compact operators      183
Operator ideal(s), Dunford — Pettis operators      307
Operator ideal(s), strictly cosingular operators      208—209
Operator ideal(s), strictly singular operators      205
Operator ideal(s), weakly compact operators      172
Operator(s)      37—38 see "Compact
Operator(s) with closed range      166—169 187—188
Operator(s), adjoint      164—170
Operator(s), almost open      79
Operator(s), ascent of      192—193
Operator(s), bounded and continuous basic equivalences      38
Operator(s), continuous in locally convex spaces      114—115
Operator(s), convergence of on dense subspaces      99
Operator(s), descent of      193
Operator(s), Dunford — Pettis      305 306—310
Operator(s), eigenvalues and eigenvectors of      198
Operator(s), extension from dense subspace      40—41 42
Operator(s), finite rank      179
Operator(s), graph of      81
Operator(s), into $l_{1}$ or on $c_{0}$       182—183
Operator(s), linear      37
Operator(s), nilpotent      197
Operator(s), noncompact with compact square      196
Operator(s), null space (kernel)      160
Operator(s), open map      24
Operator(s), properties of mill space and range      194—198
Operator(s), range of      159
Operator(s), reflexivity of $\mathcal{L}(\mathcal{X},\mathcal{X})}$       321
Operator(s), semivariation of      180
Operator(s), spectrum of      200—201
Operator(s), unconditionally converging      304—305
Operator(s), when quotient      170
Orlicz — Pettis theorem      280—282
Orlicz, W.      95 269 277 280 see
Parseval's identity      162
Pelczynski, A.      210 214 218 219 220 224 241 243 245 246 247 285 299 300 301 304 309 see
Pettis' theorem      282
Pettis, B.J.      181 269 277 280 282 333 see "Orlicz "Pettis'
Phelps, R.R.      310 313 322 see
Phillips' Lemma      270—274 275 276 277

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