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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.

Язык:

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

$c_{0}$       32
$c_{0}$ and block basic sequences      243—245
$c_{0}$ and compact sets      189
$c_{0}$ and matrix maps      96—98
$c_{0}$ and w.u.c.-series      283—284 285
$c_{0}$ not complemented in $l_{\infty}$       179 275
$c_{0}$ not complemented in $l_{\infty}$ and projections      159—161
$c_{0}$ not complemented in $l_{\infty}$ in Hilbert space or $l_{2}$       162—163
$c_{0}$ not complemented in $l_{\infty}$ in spaces with basis      242—243
$c_{0}$ not complemented in $l_{\infty}$ , finite-dimensional subspaces      161
$c_{0}$ , basis of      233
$c_{0}$ , basis of, has no boundedly complete basis      301
$c_{0}$ , basis of, nonslirinking basis for      290
$c_{0}$ , basis of, shrinking      289
$c_{0}$ , compact operators on      183
$c_{0}$ , complemented in separable spaces      163
$c_{0}$ , complemented subspaces isomorphic to $c_{0}$ in      245—246
$c_{0}$ , contained as subspace      283—285
$c_{0}$ , dense subspace      42
$c_{0}$ , dual      44—45
$c_{0}$ , Dunford — Pettis Property      306
$c_{0}$ , not a dual space      149
$c_{0}$ , not complemented in $l_{\infty}$       179 275
$c_{0}$ , one operator ideal on      184
$c_{0}$ , separable      34 71
$c_{0}$ , strictly singular operators on      202
$c_{0}$ , unit ball not weakly compact      126—127
$c_{0}$ , weakly compactly generated      176
$c_{0}$ , when all operators into are weakly compact      177—179
$L_{1}$       71 101 102—104
$l_{1}$ as dual of $c_{0}$       44—45
$l_{1}$ as dual of c      48—49
$l_{1}$ , boundedly complete basis      301
$l_{1}$ , coincidence of weak and norm sequential convergence in      276
$l_{1}$ , compact and weakly compact operators into      182—183
$l_{1}$ , complemented in bidual $ba(\mathcal{P}(\mathbb{N}))$       276
$l_{1}$ , contained in spaces with Schur property      259—260
$l_{1}$ , dual      45 46
$l_{1}$ , has Dunford — Pettis Property      305—306
$l_{1}$ , is weakly sequentially complete      276—277
$l_{1}$ , Rosenthal's criteria for selecting basic sequence equivalent to $l_{1}$ basis      259—267
$l_{1}$ , separable spaces are quotients of,      103—104
$L_{1}(\mu)$       174 179
$L_{1}(\mu)$ is weakly compactly generated      176
$L_{1}(\mu)$ , continuity of linear maps into      103
$L_{1}(\mu)$ , dense subspace of      53
$L_{1}(\mu)$ , dual of      54
$L_{1}(\mu)$ , has Dunford — Pettis Property      306
$L_{1}(\mu)$ , has no boundedly complete basis      301
$L_{1}(\mu)$ , not a dual space      149
$L_{1}(\mu)$ , unconditionally convergent series in      280—282
$L_{1}(\mu)$ , when continuous operators on are weakly compact      181
$l_{p}$ $(1 \leq p < \infty)$       29—32 36 104 see
$l_{p}$ $(1 \leq p < \infty)$ and compact sets      189
$l_{p}$ $(1 \leq p < \infty)$ , basis for      233
$l_{p}$ $(1 \leq p < \infty)$ , basis for, block basic sequences in      243—246
$l_{p}$ $(1 \leq p < \infty)$ , basis for, boundedly complete      301
$l_{p}$ $(1 \leq p < \infty)$ , basis for, shrinking      289—290
$l_{p}$ $(1 \leq p < \infty)$ , compact operators between      254—255
$l_{p}$ $(1 \leq p < \infty)$ , dense subspace      42
$l_{p}$ $(1 \leq p < \infty)$ , dual      46—48
$l_{p}$ $(1 \leq p < \infty)$ , isomorphic complemented subspaces in      245—246
$l_{p}$ $(1 \leq p < \infty)$ , one operator ideal on      184
$l_{p}$ $(1 \leq p < \infty)$ , separablility      33—34
$l_{p}$ $(1 \leq p < \infty)$ , strictly singular operators on      202
$l_{p}$ $(1 \leq p < \infty)$ , total incomparability      255 256
$L_{p}(\mu)$ $(1 \leq p < \infty)$       35—36 72 104
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , compact sets in      189—190
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , dense subspace of      42
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , dual of      54—56
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , Haar system for      234—237
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , Haar system for, is boundedly complete      301
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , Haar system for, is shrinking      290
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , is uniformly convex      336—337
$L_{p}(\mu)$ $(1 \leq p < \infty)$ , is uniformly smooth      337
$L_{\infty}$       32—33 71 87—88 101 174
$l_{\infty}$ and Grothendieck spaces      175
$l_{\infty}$ and matrix maps      96—98
$l_{\infty}$ as dual of $l_{1}$       45—46
$l_{\infty}$ , $c_{0}$ not complemented in      179 275
$l_{\infty}$ , dual      50
$l_{\infty}$ , lifting property of      82—83
$l_{\infty}$ , noncompact operator on with compact square      196
$l_{\infty}$ , nonseparability of      34
$l_{\infty}$ , representation of continuous linear operators on      274—275
$l_{\infty}$ , separable subspaces not complemented      275
$L_{\infty}(\mu)$       36
$L_{\infty}(\mu)$ , dense subspace of      42
$L_{\infty}(\mu)$ , has Dunford — Pettis property      306
$L_{\infty}(\mu)$ , pointwise limit of weakly compact operators in is weakly compact      179
$L_{\infty}(\mu)$ , weakly compact operators on      181
$\mathcal{B}(B_{0},\mathcal{Y})$       105
$\mathcal{B}(\mathcal{K})$       151—152
$\mathcal{B}(\Omega)$       25—26 32 33
$\mathcal{B}(\Omega)$ as $C(\mathcal{K})$ -space      26
$\mathcal{B}(\Omega)$ as dual of $C(\mathcal{K})$ -space      59—60
$\mathcal{B}(\Omega, \Sigma)$       50—51
$\mathcal{B}(\Omega, \Sigma)$ , dense subspace      50
$\mathcal{C}(\mathcal{K})$       20—21 26 57
$\mathcal{C}(\mathcal{K})$ and separability      157—158
$\mathcal{C}(\mathcal{K})$ and Stone — Weierstrass Theorem      151—153
$\mathcal{C}(\mathcal{K})$ , dual      59—60
$\mathcal{C}(\mathcal{K})$ , extreme points in dual      148—149
$\mathcal{C}(\mathcal{K})$ , has Dunford — Pettls Property      305
$\mathcal{C}(\mathcal{K})$ , weakly compact operators on      180—181 189 310
$\mathcal{C}(\mathcal{K})$ , weakly compact sets      189
$\mathcal{C}(\mathcal{K})$ , when a Grothendieck space      175
$\mathcal{C}(\mathcal{K})$ , when weakly sequentially complete      157
$\mathcal{C}[0,1]$       19—20 26 82 104
$\mathcal{C}[0,1]$ and compact operators      185—186 189
$\mathcal{C}[0,1]$ , all separable spaces imbed in      142
$\mathcal{C}[0,1]$ , dense subspace,      42
$\mathcal{C}[0,1]$ , no boundedly complete basis      301
$\mathcal{C}[0,1]$ , Schauder basis for      233—234
$\mathcal{C}[0,1]$ , separable      33
$\mathcal{C}_{0}(\mathcal{K})$       26—27 33
$\mathcal{C}_{c}(\mathcal{K})$       57—59
$\mathcal{V}^{\infty}(\mu, \mathcal{X})$       51-54
$\mathcal{V}^{\infty}(\mu, \mathcal{X})$ as dual of $L_{1}(\mu)$       54
$\mathcal{V}^{\infty}(\mu, \mathcal{X})$ , realized as $\mathcal{L}(L_{1}(\mu),\mathcal{X})$       53
$\sigma (\mathcal{F},\mathcal{G})$ -topology      119—121
$\sigma \mathcal{(F,G)}$ -topology, dual with respect to      119—120
${ba}(\mathcal{P}_{N})$       50—51 270
${ba}(\mathcal{P}_{N})$ as dual of $l_{\infty}$       50—51
Absolutely summable series      27
Absorbing set(s) (radial at $\theta$ )      109 110—111
Adjoint operator(s)      164 165—170
Adjoint operator(s) of strictly singular and cosingular operators      209—210
Adjoint operator(s), basic properties      166
Adjoint operator(s), closed and injective range implies operator surjective      168—169
Adjoint operator(s), closed range characterization      166—170
Adjoint operator(s), defining an embedding of $\mathcal{L(X,Y)}$ into $\mathcal{L(Y^{*},X^{*})}$       165
Adjoint operator(s), weak*-continuity of      166
Affine function(s)      215
Alaoglu, L.      124 125 see
Amir — Lindenstrauss Theorem      177 216
Amir, D.      176 219 see
Anderson, R.D.      218
Ando.T.      175
Approximation problem      184 219 220
Approximation property      218 321
Arzela — Ascoli theorem      182 186
Assent of an operator      192—193
Baire category theorem      76 78 124 221
Baire measure(s)      57
Baire sets      57
Banach Closed Graph Theorem      see "Closed Graph Theorem"
Banach Homomorphism Theorem      see "Open-Mapping Theorem"
Banach Isomorphism Theorem      63 80—81 222 226 291
Banach limit(s)      87—94

Banach space(s)      18
Banach space(s) with Frechet differentiable norm      328—330
Banach space(s) with Gateaux differentiable norm      323—325
Banach space(s) with uniformly Frechet differentiable norm      330—333
Banach space(s), always contain subspace with basis      252
Banach space(s), always weak* dense in bidual      127—128
Banach space(s), criteria for containing subspace isomorphic to $c_{0}$       284—285
Banach space(s), direct sums of      211—213
Banach space(s), grothendieck space      175 177 179 184 309
Banach space(s), nonreflexivity characterization      317—319
Banach space(s), norm achieving functionals on unit ball are dense      313—314
Banach space(s), reflexive exactly when closed subspaces with basis are      297
Banach space(s), smooth      323—325
Banach space(s), somewhat reflexive      257
Banach space(s), strictly convex (rotund)      325—328
Banach space(s), subreflexive      313
Banach space(s), uniformly convex      330—336
Banach space(s), uniformly smooth      330—336
Banach space(s), very smooth      330
Banach space(s), weak* sequential closure is norm closed in bidual      128—129
Banach space(s), weakly compactly generated      176
Banach space(s), weakly complete implies finite-dimensional      157
Banach space(s), weakly conditionally compact      254
Banach space(s), when isomorphic to Hilbert space      60—62
Banach — Alaoglu theorem      125 126 127 133 137 140 142 144 148 149 150 153 168 171 173 215 320 324
Banach — Alaoglu Theorem with respect to dual spaces      126
Banach — Mazur theorem      142 179 251 252 256 301
Banach — Steinhaus theorem      63 76—78 82 94 95 98 101 103 105 106 130 144 159 178 222 224 227 228 230 294 295 300
Banach's theorem      168 169 252 297
Banach's Weak Basis Theorem      222 229 230 239 296
Banach, Stefan      2 27 63 64 75 76 77 78 81 87 88 89 90 92 95 124 125 127 140 166 169 219 220 245 252 see "Banach "Banach "Banach "Banach's "Banach "Hahn
Bartle — Dunford — Schwartz Theorem      180 181
Bartle, R.G.      180 see
Basic selection principles      see "Selection principles"
Basic sequence(s)      227—229
Basic sequence(s) and reflexivity      300—301
Basic sequence(s) in dual spaces      232
Basic sequence(s) in weakly null sequences      251—252
Basic sequence(s), Bessaga — Pelczynski criteria for selecting basic subsequence      249—251
Basic sequence(s), block      243—245
Basic sequence(s), equivalence of      240
Basic sequence(s), existence of with constant near one      252—252 253
Basic sequence(s), in weakly convergent sequences are bounded away from $\theta$       251—252
Basic sequence(s), Mazur's criteria for selecting basic subsequence      247—249
Basic sequence(s), Rosenthal criteria for selecting basic subsequences and $l_{1}$       259—267
Basic sequence(s), when equivalent to basis      241—242
Basic sequence(s), when equivalent to unit vector basis of $c_{0}$       244
Basis constant      232
Basis constant, existence of basic sequences with constant near one      252—253
Basis problem      219
Basis(es)      217 see "Shrinking "Boundedly
Basis(es) and complementation      242—243
Basis(es) and dense subsets      242
Basis(es) and subsequences      228—229
Basis(es) for James' space $\mathbf{J}$       237—238
Basis(es) in reflexive spaces      286 297
Basis(es), associated coefficient functionals      220 225—226
Basis(es), Banach spaces always have subspace with      252
Basis(es), boundedly complete      294
Basis(es), characterization of for biorthogonal system      228—229
Basis(es), equivalence of weak and Schauder      222—222 230
Basis(es), equivalent bases      238—242
Basis(es), Haar system for $L_{p}([0,1])$       234—237
Basis(es), Hamel basis uncountable      221—222
Basis(es), internal characterization of      230—232
Basis(es), monotone      232 234
Basis(es), perturbation of      241—242
Basis(es), Schauder      220
Basis(es), Schauder system in C([0,1])      233—234
Basis(es), shrinking      286
Basis(es), weak      220
Basis(es), weak*-Schauder      297 297
Bessaga — Pelczynski Selection Principle      220 249—251 252 254 257 269 282 285 304
Bessaga, C.      218 220 224 242 247 285 301 see
Best approximations      149—151
Bidual      72
Biorthogonal system(s)      226—229
Biorthogonal system(s) and expansion operators      226
Biorthogonal system(s), when yields basis or basic sequence      228 229
Bishop — Phelps theorem      270 310 311 312 313—314 330
Bishop, E.      310 313 322 see
Block basic sequence(s)      243—246
Block basic sequence(s) and shrinking bases      288—289
Block basic sequence(s) in $c_{0}$ and $l_{p}$       243—245
Bochner, S.      180 181
Bohnenblust, H.F.      83
Bounded approximation property      218 219—220
Bounded multiplier Cauchy (convergent) series      277 see
Boundedly complete basis(es)      294
Boundedly complete basis(es) and shrinking yields reflexivity      297—299
Boundedly complete basis(es), examples      301
Boundedly complete basis(es), existence of guarantees being dual space      295—296
Boundedly complete basis(es), relationship with shrinking basis      294—295
Bourbaki, N.      127
Brooks, J.K.      77
C      32 89 100
c and matrix maps      96—99
c, dual      48—49
c, not isometrically isomorphic to $c_{0}$       49
c, separable      34 71
Canonical injection j      see "Natural injection"
Cantor set $\mathbf{C}$       140—142 338—339
Cartesian produces(s)      see "Produces"
Choquet, G.      150
Circled (balanced) set(s)      108 109—111
Circled hull $c(\cdot)$       109
Clarkson, J.A.      270 334 336
Closed graph theorem      63 81—82 103 105 106 161 165 305
Coefficient functionals      220
Collins, H.S.      130
Compact operator(s)      181 182—186
Compact operator(s) and C([0,1])      185—186 189
Compact operator(s) and the Riesz — Schauder theory      186—201
Compact operator(s) as operator ideal      183
Compact operator(s) on finite-dimensional spaces      182
Compact operator(s) on weakly sequentially compact spaces      254
Compact operator(s), between $l_{p}$ -spaces      254—255
Compact operator(s), exactly when adjoint compact      182
Compact operator(s), exactly when map bounded nets to convergent nets      184
Compact operator(s), is strictly cosingular      206
Compact operator(s), matrix form on $l_{2}$       184—185
Compact operator(s), pointwise limit of on Grothendieck spaces      184
Compact operator(s), properties of      184—186
Compact operator(s), uniform limit of      184
Compactness and extreme points      146—148
Compactness, examples      189—190
Compactness, sequential, and conditionally countable      129 130—136
Comparable norms      see "Norm(s)"
Complemented subspace(s)      161
Complemented subspace(s) in $c_{0}$ and $l_{p}$       243—246
Complemented subspace(s), $c_{0}$ is in any separable space      163
Complemented subspaces problem      161
Completeness      18
Completeness, exactly when absolutely summable sequences are summable      27—28
Continuity of addition and scalar multiplication      21—22
Continuity, weak and weak*      158—159
Convex cone      311 312—313
Convex conjugates      29
Convex hull $co(\cdot)$       108—109
Convex set(s)      108 109—113
Convex set(s) and extreme points      144—146
Convex set(s) in locally convex spaces      121
Convex set(s), coincidence of weak and norm closure      122
Convex set(s), norm achieving functionals on      313—319
Convex set(s), support points and functionals for      311—313
Convex set(s), when weak* closed in dual      142—143
Conway, J.B.      87
Coordinate (projection) maps      5
Dabokov, R.      306
Daniell integral      11—16 58

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