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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.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1965

Количество страниц: 295

Добавлена в каталог: 20.02.2013

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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       146 Dual of       93 Dual of       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 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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