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Goffman C., Pedrick G. — First course in functional analysis
Goffman C., Pedrick G. — First course in functional analysis

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Название: First course in functional analysis

Авторы: Goffman C., Pedrick G.

Аннотация:

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.
The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that subject. However, the general concept of functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.
In modern introductory texts to functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, and does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1965

Количество страниц: 295

Добавлена в каталог: 20.02.2013

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Absolutely continuous measure      132
Adjoint of a transformation      84
Algebra      33
Algebra, Banach      248
Algebraic dual      52
Almost everywhere      113
Approximate identity      249 269
Arc length      40
Arzela — Ascoli theorem      28
Banach algebra      248
Banach lattice      236
Banach limits      64
Banach space      71
Base for a topology      206
Basis, Hamel      55
Basis, orthogonal      177
Basis, Schauder      101
Bergman kernel      188
Bessel’s inequality      176
Bounded, functional      72 233
Bounded, set      211
Bounded, totally      25
Bounded, transformation      72
Bounded, uniformly      28
Category      21
Cauchy sequence      11
Cauchy — Schwarz inequality      4 6 165
Chain      53
CHARACTER      270
Characteristic number      85
Closed graph theorem      98
Closed set      9 208
Closed transformation      97
Closure of a set      9 208
Coarser topology      206
Compact      24 208
Compact transformation      83
Compact, locally      251
Complete metric space      11
Complete set      173 191
Completion      20
Completion, functional      185
Connected      28
Continuity      27 207
Continuity, absolute      132
Continuity, equi      28
Continuity, semi      38
Continuity, uniform      27
Contraction      17
Convergence in measure      118
Convergence, exercise      2 8
Convergence, field      231
Convergence, weak      86 202
Convergent sequence      8
Convex      7 58 211
Convex, locally      217
Curve      39
Deficiency      57
Dense      9
Dense, everywhere      9
Dense, nowhere      21
DIMENSION      56 183
Directed set      246
Directed set, exercise      10 1
Directed sum      51 100
Dominated Convergence Theorem      127
Dual of $L_{1}$      146
Dual of $l_{p}$      93
Dual of $L_{p}, p>1$      142
Dual of C[a, b]      94
Dual, algebraic      52
Dual, space      73
Egoroffs theorem      118
Entire functions      220
Equicontinuity      28
Ergodic theorem, individual      148
Ergodic theorem, mean      172
Everywhere dense      9
Fatou theorem      126
Field, convergence      231
Field, normed      257
Finer topology      206
First category      21
FK space      224
Fourier expansion      175—77
Frechet space      224
Fubini’s Theorem      161
Fubini’s theorem, exercise      6 4
Functional      52
Functional completion      185
Functional, bounded      (see Bounded transformation)
Functional, linear      52
Functional, positive      233
Graph of a transformation      98
Group, locally compact      251
Group, ordered      59
Group, topological      251
Haar functions      194
Haar measure      252
Hahn — Banach theorem      60 62 73
Hamel basis      55
Hilbert space      166
Holder inequality      2 6 136
Hyperplane      57
Ideal      254
Ideal, maximal      255
Identity, approximate      249
Independent, linearly      55
Individual ergodic theorem      148
Inner product      165
Inner product space      166
Integrable function      124
integral      120—24 128
Interior point      9
Involution      265
Isometry      20
Kernel, Bergman      188
Kernel, reproducing      186
Kothe space      239
Lattice, Banach      236
Lattice, vector      233
Lebesgue      120—24
Lebesgue integral      120—24
Lebesgue measure      115
Limit point      9
Linear, functional      52
Linear, independence      55
Linear, space      (see Vector space)
Linear, transformation      52
Lipschitz condition      17
Locally compact group      251
Locally convex space      217
Mapping      (see Transformation)
Maximal, element      53
Maximal, ideal      255
Mean ergodic theorem      172
Measurable function      115 212
Measurable set      110—14
Measure      109—15
Measure, absolutely continuous      132
Measure, convergence in      118
Measure, Haar      252
Measure, Lebesgue      115
Measure, Radon      237
Measure, signed      129
Measure, singular      134
Measure, totally a finite      115
Measure, totally finite      114
Metric      1
Metric space      1
Metric, uniform      8
Metrizable      207 219
Miintz’ theorem      181
Minkowski inequality      3 5
Neighborhood      9
Norm      71
Norm of a transformation      72
Norm, semi      71 217
Normable space      211
Normal family      35
Normed field      257
Nowhere dense set      21
Null space      57
Open mapping theorem      98
Open set      9 206
Operator      (see Transformation)
Operator, positive      102
Ordered set      232
Ordered set, partially      53
Orthogonal      167
Orthogonal, basis      177
Orthogonal, set      173
Orthonormal set      173
Parseval’s formula      177
Partially ordered set      53
Positive functional      233
Positive operator      102
Product, inner      165
Product, topology      208
Projection      100 170
Projection theorem      170
Rademacher system      197
Radical      265
Radon measure      237
Radon — Nikodym theorem      132
Reflexive space      91
Regular ideal      266
Regular method      87 228
Reproducing kernel      186
Riemann — Stieltjes      95
Riesz representation theorem      96
Riesz — Fischer theorem      184
Schauder basis      101
Schwarz inequality      4 6 165
Second category      21
Semi-continuity      38
Semi-simple      265
Seminorm      71 217
Separable space      9 183
Sgn z      93
Signed measure      129
Singular measure      134
Space of entire functions      220
Space of measurable functions      212
Space, Banach      71
Space, compact      208
Space, complete metric      11
Space, linear      (see Vector space)
Space, locally convex      217
Space, m      75 (see also $l_{p}</a></span> <span class=subjpages><a href=p=\infty$"/>)
Space, metric      1
Space, M[a, b]      6
Space, normable      211
Space, reflexive      91
Space, separable      9 183
Space, topological      206
Space, topological vector      210
Space, vector      50
Spectrum      263
Sphere      8
Stone — Weierstrass theorem      34
Subbase for topology      206
subspace      2 52
Summable function      123
Summand, direct      51 100
Topological group      251
Topological space      206
Topological vector space      210
topology      206
Topology, base for      206
Topology, coarser      206
Topology, finer      206
Topology, product      208
Topology, subbase for      206
Topology, weak      209
Total set      57 86 214
Total variation      131
Totally a finite measure      115
Totally bounded      25
Totally finite measure      114
Transformation, adjoint      84
Transformation, bounded      72
Transformation, closed      97
Transformation, compact      83
Transformation, graph of a      98
Transformation, linear      152
Transformation, positive      102
Tychonoff theorem      208
Uniform boundedness      28
Uniform boundedness, principle of      76
Uniform continuity      27
Uniform metric      8
Variation, total      131
Vector lattice      233
Vector space      50
Vector space, topological      210
Walsh system      200
Weak convergence      86 202
Weak convergence, exercise      2 8
Weak topology      209
Weierstrass approximation theorem      32
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