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

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Ќазвание: Functional Analysis: An Introduction to Banach Space Theory

јвтор: Morrison T.M.


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 readerТs 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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 2000

 оличество страниц: 359

ƒобавлена в каталог: 26.10.2010

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ѕредметный указатель
Phillips's theorem      275
Phillips, R.S.      175 269 270 275 see "Phillip's
Pietsch operator ideal      see "Operator ideal"
Pietsch, A.      171 see
Pitt Ч Rosenthal Theorem      254Ч255
Pitt, H.R.      254 see
Principal of Uniform Boundedness      see "Banach Ч Steinhaus Theorem"
Product(s) (Cartesian) and the product topology      4Ч8
Product(s) (Cartesian) and the product topology, continuity in      6
Product(s) (Cartesian) and the product topology, fundamental system of neighborhoods      5Ч6
Product(s) (Cartesian) and the product topology, local base      5
Product(s) (Cartesian) and the product topology, tie to weak topology      7Ч8
Projection(s)      159Ч163
Projection(s) and complemented subspaces      160Ч161
Projection(s) and direct sums      212Ч213
Pseudonorm      see "Seminorm"
Ptak, V.      130
Quotient(s)      23Ч25
Quotient(s), dual of space      51
Quotient(s), exactly when adjoint is isomorphism      170
Quotient(s), map      24 78
Quotient(s), properties of,      24Ч25
Quotient(s), space      24
Radon Ч Nikodym theorem      54 55 103 277 321 334
Range of operator      159
Range of operator and complementation      195Ч196
Range of operator, closed, implications of      166Ч170
Range of operator, dense      168
Range of operator, properties of      194Ч198
Range of operator, reformulation of range when closed      169
Reflexivity      72
Reflexivity and basic sequences      300Ч301
Reflexivity and factoring weakly compact operators      214Ч216
Reflexivity and Frechet differentiability in dual      330
Reflexivity and nearest points for convex set      326Ч327
Reflexivity and norm achieving functionals      317Ч319
Reflexivity and smootlmess, support mappings      330
Reflexivity and strict convexity, smoothness      327 329
Reflexivity and unconditionally convergent series      280Ч282
Reflexivity and weakly compact sets      216
Reflexivity for $\mathcal{L}(\mathcal{X},\mathcal{Y})$      321
Reflexivity, basis(es) for dual      297
Reflexivity, basis(es), characterization of      300Ч301
Reflexivity, basis(es), coefficient functionals basis for dual      286
Reflexivity, completeness in      75
Reflexivity, exactly when closed subspaces are, and exactly when bounded sequences have weak convergent subsequences      136Ч136 137
Reflexivity, exactly when closed subspaces with bases are      299Ч300
Reflexivity, if dual uniformly smooth      334
Reflexivity, is uniformly convex      333Ч334
Retherford, J.R.      290
Riesz representation theorem      54Ч56
Riesz Ч Kakutani Representation Theorem      59Ч60 144 180
Riesz Ч Schauder theory      155 184 186Ч201
Riesz's Lemma      10Ч11
Riesz, F.      2 10 104 181 187 see "Riesz "Riesz "Riesz
Rosenthal Selection Principle      259Ч267
Rosenthal's Theorem      259Ч267 283 300
Rosenthal, A.      254 see
Rosenthal, H.P.      219 220 256 258 260 261 266 see "Rosenthal's
Rotundity      see "Strict convexity"
Royden, H.L.      12 35
Rutman, M.A.      242 see
Saks, S.      76 277
Schaefer, H.H.      24
Schauder basis(es)      see "Basis(es)"
Schauder system for C([0,1])      233Ч234
Schauder's theorem      182 183 184 193
Schauder, J.      78 182 187 217 233 235 see 1])" "Schauder's
Schur property      182 259
Schur Property, $l_{1}$ has      276
Schur Property, closed subspace contains copy of $l_{1}$      259Ч260
Schur space(s)      254
Schur's theorem      269 276 305
Schur, J.      276 see "Schur "Schur's
Schwartz, J.T.      142 156 187
Schwartz, L.      130
Seever, G.L.      175
Selection principles, Bessaga Ч Pelczynski      249Ч251
Selection principles, Mazur      247Ч248
Selection principles, Rosenthal      259Ч267
Seminorm      24 110 118Ч119
Separability      33
Separability as quotients of $l_{1}$      103Ч104
Separability in $\mathcal{C(K)}$ or $\mathcal{C}(\mathcal{K})^{*}$      157Ч158
Separability, all linearly isometric to subspace of C([0,1])      160
Separability, are weakly compactly generated      176
Separability, bounded sequences have weak* convergent subsequences in dual      140
Separability, dual is implying space is      71
Separability, exactly when $U_{x^{*}}$ weak*-metrizable      137Ч139
Series, coincidence of weak and norm subseries convergence      280Ч282
Series, equivalence of unconditionally Cauchy (convergent), unordered Cauchy (convergent), subseries Cauchy (convergent), and bounded multiplie Cauchy (convergent)      277Ч280
Series, weakly unconditionally convergent,      282Ч283
Set(s), $\sigma$-bounded      57
Set(s), $\sigma$-compact      57
Set(s), absorbing      109
Set(s), annihilator of      151
Set(s), Baire      57
Set(s), bounded      37
Set(s), circled (balanced)      108
Set(s), closed linear span $[\cdot]$      71
Set(s), closure (norm)      23
Set(s), convex      108
Set(s), distance from point to      68
Set(s), norming      129 247
Set(s), polar of      143
Set(s), radial at x      109
Set(s), total      129
Set(s), weak* bounded      157
Shrinking basis(es),      286
Shrinking basis(es), and boundedly complete yields reflexivity      297Ч299
Shrinking basis(es), boundedly complete basis, relationship with      294Ч295
Shrinking basis(es), equivalences and bounded sequences      288Ч289
Shrinking basis(es), exactly when coefficient functionals are basis for dual      287Ч288
Shrinking basis(es), examples      289Ч290
Shrinking basis(es), generates an equivalent norm in dual      290Ч292
Shrinking basis(es), of James' space $\mathbf{J}$      290
Silverman Ч Toepliz Summability Theorem      63 100Ч101
Silverman, L.L.      63 100 see
Singer, I.      150 246 296 300
Smoothness      323 324Ч325
Smoothness and Gateaux differentiability      324Ч325
Smoothness and strict convexity      323
Smoothness, reflexivity and support mappings      330
Smoothness, very smooth      330
Smulian, V.L.      127 130 131 142 328 see "Krein
Sobczyck, A.      83 163
Somewhat reflexive      257
Somewhat reflexive, James' space is      257Ч258
Spectral theory      198Ч210
Spectral theory, spectrum of an operator      200Ч201
Spence, L.E.      8
Steinhaus, H.      75 76 77 see
Stone Ч Cech compactification      26
Stone Ч Weierstrass theorem      149 151 152Ч153
Stone's theorem      15Ч16 58
Stone, M.H.      14 15 see "Stone's
Strict convexity (rotundity)      325 326Ч328
Strict convexity (rotundity) and reflexivity      327
Strict convexity (rotundity) and smoothness      327
Strict convexity (rotundity), metric characterization of      325Ч326
Strictly cosingular operator(s)      206 207Ч210
Strictly cosingular operator(s) and compact operators      206
Strictly cosingular operator(s) and the Dunford-Pettis property      309
Strictly cosingular operator(s) and weakly compact      206
Strictly cosingular operator(s), adjoint of      209Ч210
Strictly cosingular operator(s), operator ideal      207Ч209
Strictly singular operator(s)      201 202Ч210
Strictly singular operator(s) and compact operators      202
Strictly singular operator(s) and weakly compact operators      209Ч210
Strictly singular operator(s) on $c_{0}$ or $l_{p}$      202
Strictly singular operator(s), adjoint of      209
Strictly singular operator(s), characterization of      205
Strictly singular operator(s), if Dunford Ч Pettis      307Ч308
Strictly singular operator(s), implies unconditionally converging      304Ч305
Strictly singular operator(s), operator ideal      205Ч206
Strictly singular operator(s), when largest two-sided ideal      308Ч309
Subseries Cauchy (convergent)      277 see
Subseries Cauchy (convergent), coincidence of weak and norm      280Ч282
Subspace(s) and reflexivity      136Ч137 299Ч300
Subspace(s) of finite codimension      207
Subspace(s) of finite deficiency and compact operators      202Ч204
Subspace(s), always exist with basis      252
Subspace(s), closure      70
Subspace(s), codimension of      292
Subspace(s), dual of      51
Subspace(s), extension of linear functionals on      68
Subspace(s), is subspace      22
Subspace(s), maximal closed subspaces are isomorphic      299
Subspace(s), separation from points outside      69
Subspace(s), when dense      70
Subspace(s), when dense, dense examples      42 50 53
Subspace(s), when weak* closed in dual      143
Summable sequences      27
Support mapping(s) on unit sphere      322Ч323
Support mapping(s) on unit sphere and reflexivity      330
Support mapping(s) on unit sphere and smoothness, Gateaux differentiability      324Ч325
Support mapping(s) on unit sphere and uniform smoothness, uniform convexity of dual      332Ч333
Support mapping(s) on unit sphere, Frechet differentiability      329
Supporting set(s)      146
Supporting set(s) of a measure      152
Taylor, A.E.      11 187
Toeplitz, O.      63 100 see
Topological dual space(s)      see "Dual space(s)"
Topological vector space(s)      21 112
Total set(s)      70 129
Total set(s), when countable in dual      129
Totally incomparable spaces      255
Totally incomparable spaces, $l_{p}$-spaces      256
Totally incomparable spaces, sum of always closed      256Ч257
Tsing, Nam-Kiu      221
Tychonoffs theorem      125
Tzafriri, L.      161 162 163 252
Uhl, J.J.      54 180
Unconditionally Cauchy (convergent) series      211 see
Unconditionally Cauchy (convergent) series and uniform convexity      338Ч339
Unconditionally converging operator(s)      304
Unconditionally converging operator(s), examples      305Ч306
Unconditionally converging operator(s), if strictly singular      304Ч305
Uniformly convex      331 332Ч339
Uniformly convex and modulus of convexity      337
Uniformly convex and unconditionally convergent series      338Ч339
Uniformly convex of $L_{p}(\mu)$-spaces      336Ч337
Uniformly convex, characterization      331 332Ч333
Uniformly convex, dual characterization      332Ч333
Uniformly convex, is reflexive      333Ч334
Uniformly Frechet differentiable norm      see "Differentiability"
Uniformly smooth      330 331
Uniformly smooth, characterization      332Ч333
Uniformly smooth, if dual is then reflexive      334
Unit ball $U_{\mathcal{X}}$      36
Unit ball $U_{\mathcal{X}}$ and extreme points in dual      150
Unit ball $U_{\mathcal{X}}$, compact exactly when finite-dimensional      133
Unit ball $U_{\mathcal{X}}$, dual ball weak*-metrizable exactly when space separable      137Ч139
Unit ball $U_{\mathcal{X}}$, nonweak compactness of      125Ч126
Unit ball $U_{\mathcal{X}}$, norm achieving functionals are dense      313Ч314
Unit ball $U_{\mathcal{X}}$, support mappings on      322Ч323
Unit ball $U_{\mathcal{X}}$, weak* compactness of dual ball      125
Unit ball $U_{\mathcal{X}}$, weak* denseness in bidual unit ball      127Ч128
Unit ball $U_{\mathcal{X}}$, weakly metrizable exactly when dual separable,      137Ч139
Unit sphere $S_{\mathcal{X}}$      37
Unit sphere $S_{\mathcal{X}}$, nearness to closed subspaces with elements of $S_{\mathcal{X}}$      10Ч11
Unordered Cauchy (convergent) series      277 see
Urysohn's lemma      148
Vector lattice(s)      12
Vector measure(s)      53Ч54
Vector measure(s), coincidence of weak and strong      282
Veech, W.A.      163
Very smooth      330
Vitali Ч Hahn Ч Saks Theorem      277
Vladimorskii, W.      207
Weak basis(es)      see "Basis(es)"
Weak compactness and norm achieving functionals      319Ч319 321
Weak compactness of unit ball characterizes reflexivity      127
Weak compactness, affinely homeomorphic to subset of reflexive space      216
Weak compactness, implies closed convex hull is weakly compact      143Ч144
Weak compactness, is closed and bounded      126 130
Weak compactness, metrizable in separable spaces      131
Weak topology      117Ч121 122 123 124
Weak topology, continuity and boundedness in,      158Ч159
Weak topology, convergence of nets      123
Weak topology, induced by a collection of functions      7Ч8
Weak topology, nonmetrizability of      123Ч124
Weak topology, open sets in      123
Weak topology, relativized      131Ч132
Weak topology, sets are weak bounded exactly when norm bounded      130
Weak topology, weakly compact sets      126
Weak* compactness of unit ball      125
Weak* topology      123 124
Weak* topology, continuity and boundedness in      158Ч159
Weak* topology, convergence of nets      124
Weak* topology, nonmetrizability of      124
Weak* topology, weak* bounded exactly when bounded      157
Weakly compact operator(s)      170 171Ч181
Weakly compact operator(s) and strictly singular operators      309
Weakly compact operator(s) in and on special spaces      176Ч181
Weakly compact operator(s) on $\mathcal{C}(\mathcal{K})$-spaces      310
Weakly compact operator(s), characterization of      215
Weakly compact operator(s), exactly when $T^{**}(\mathcal{X}^{**})\subseteq j(\mathcal{Y})$      171
Weakly compact operator(s), exactly when adjoint is weak*-weak continuous      172Ч173
Weakly compact operator(s), exactly when adjoint is weakly compact      173
Weakly compact operator(s), examples of nonweakly compact      174
Weakly compact operator(s), operator ideal      172
Weakly compact operator(s), some properties of      174
Weakly compact operator(s), when Dunford Ч Pettis      306Ч307
Weakly compactly generated space(s)      176 177
Weakly compactly generated space(s), characterization of      216
Weakly sequentially compact      254
Weakly sequentially compact, compact operators on      254
Weakly sequentially complete      157 282
Weakly sequentially complete, $l_{1}$ is      276Ч277
Weakly sequentially complete, is reflexive or has subspace isomorphic to $l_{1}$      259
Weakly sequentially complete, unconditional convergence of series in      280Ч282
Weakly subseries convergent series have subsequence converging to zero in norm      281
Weakly unconditionally convergent series (w.u.c. series)      283Ч285
Weakly unconditionally convergent series (w.u.c. series), basic sequence equivalent to unit vector basis of $c_{0}$      284Ч285
Weierstrass' theorem      33 42
Weierstrass, K.      see "Weierstrass' Th Stone Ч Weierstrass Theorem"
Weil, A.      189
Whitley, R.J.      130
Willard, S.      26 140 265
Yosida, K.      187
Zippin, M.      219
Zorn's lemma      8 312
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