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Saxe K. — Beginning functional analysis
Saxe K. — Beginning functional analysis

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Название: Beginning functional analysis

Автор: Saxe K.

Аннотация:

The unifying approach of functional analysis is to view functions as points in some abstract vector space and the differential and integral operators relating these points as linear transformations on these spaces. The author presents the basics of functional analysis with attention paid to both expository style and technical detail, while getting to interesting results as quickly as possible. The book is accessible to students who have completed first courses in linear algebra and real analysis. Topics are developed in their historical context, with accounts of the past ¿ including biographies ¿ appearing throughout the text. The book offers suggestions and references for further study, and many exercises. Karen Saxe is Associate Professor of Mathematics at Macalester College in St. Paul, Minnesota. She received her Ph.D. from the University of Oregon. Before joining the faculty at Macalester, she held a two-year FIPSE post-doctoral position at St. Olaf College in Northfield, Minnesota. She currently serves on the editorial board of the MAA's College Mathematics Journal. This is her first book.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 197

Добавлена в каталог: 14.02.2006

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A^{c}$      185
$B_{r}(x)$      16
$c_{0}$      8
$E^{o}$      16
$f_{+}$      45
$f_{-}$      45
$\mathcal{B}(X)$      96
$\mathcal{B}(X, Y)$      96
$\mathfrak{L}(\mathbb{R}^{n}, m)$      47
$\textit{l}^{1}$      9
$\textit{l}^{1}(\mathbb{Z})$      132
$\textit{l}^{p}$      9
$\textit{l}^{\infty}$      8
$\textit{l}^{\infty}(X, \mu)$      61
Additive set function      36
Algebra      99
Almost everywhere      48
Analytic Banach space valued function      103
Analytic function      103
Approximation problem      110 123
Arzela, Cesare      12 19
Ascoli — Arzell theorem      17 20
Ascoli, Giulio      12 19
Axiom of Choice      154 159
Baire category theorem      148
Baire, Rene      43
Banach algebra      99
Banach contraction mapping principle      161
Banach space      23
Banach — Steinhaus theorem      153
Banach — Taraki paradox      28
Banach, Stefan      1 28
Band-limited signal      172
Basis      11
Basis problem      110 123
Bell Labs      172
Bernoulli sequence      33
Bernoulli, Daniel      75
Bernoulli, James      33
Bernstein polynomial      136
Bernstein, Sergei      136
Berthollet, Claude Louis      80
Bessel, Friedrich      82
Bessel’s inequality      82 84
Birkhoff, George David      146
Bishop, Errett      144
Bishop’s theorem      144
Bohr, Niels      167
Borel sets      42 58
Borel, Emile      18 43
Boundary of a set      158
Bounded linear operator      95
Bounded set      16
Brouwer, Luitzen      160
Bunyakovskii, Viktor      7
C([a,b])      9 136
Cantor set      30 34
Cantor, Georg      12
Cauchy sequence in a metric spucc      23
Cauchy — Schwarz inequality      7
Cauchy, Augustin Louis      7 32
Cesaro operator      129
Characteristic function      45
Clopen set      29
Closed set in a metric space      16
Closure of a set      16
Compact linear operator      107
Compact set      16
Compact support, function with      73
Complement of a set      185
Complete metric space      23
Complete orthonormal sequence      77
Complete subset in a metric space      23
Complex conjugate of a complex number      183
Composition operator      129
Connected set      158
Continuous function on a metric space      5
Contraction mapping      161
Convergence in a metric space      5
Convergence in mean      86
Convolution      132
Countable set      186
Countably additive set function      36
Countably subadditive set function      38
Counting measure      58
Cover of sets      16
de Branges, Louis      144
de Morgan’s laws      186
Dense set      16
Diagonal operator      102
Diagonal operator, characterization of compactness of      108
Dirac, Paul      167
Discrete metric      4
Discrete probability space      34
Disjoint sets      185
Dual space      154
Dunham, William      150
Einstein, Albert      27
Enflo, Per      110 112 122
Equicontinuous function      19
Equivalent norms      30
Equivalent sets      186
Essentially bounded function      61
Euclidean norm      5
Eudoxos      32
Fatou, Pierre      52
Fatou’s Lemma      52 64
Fields Medals      26
Finite-dimensional      11
Finite-rank operator      110
Finitely subadditive set function      69
First category set      147
Fourier coefficients      82
Fourier series      82
Fourier transform      167
Fourier, Joseph      4 75 79
Frechet, Maurice      1 11 33
Fredholm operator of the first kind      94
Fredholm operator of the second kind      94
Fredholm, Erik Ivar      12 68 94
Fundamental theorem of algebra      104
Fundamental theorem of calculus      56
Grabiner, Sandy      145
Gram — Schmidt process      77
Haar functions      78
Haar, Alfred      79
Hadamard, Jacques      12
Hahn — Banach theorem (real case)      155
Hahn, Hans      151
Hahn-Banach theorem (complex case)      156
Hamel basis      11
Hausdorff, Felix      13
Heine — Borel theorem      17 18
Heine, Eduard      18
Heisenberg, Werner      167
Helly, Eduard      154
Hermite functions      78
Hermite, Charles      78 114
Hermitian operator      114
Hermitian symmetry property of an inner product      6
Hilbert space      23
Hilbert, David, I      24 33 128
Hilbert’s      23
Hilden, Hugh      120
Hoelder conjugate      60
Hoelder, Otto Ludwig      60
Hoelder’s inequality      60
Ideal      165
Imaginary part of a complex number      183
Index set      185
Infimum of a set      9
Infinite-dimensional      11
Inner product      6
Inner product space      6
Institut d’Egypte      80
Int(E)      16
Interior of a set      16
Interior point      16
International Congress of Mathematicians      26
Intersection of sets      185
Invariant subspace      112
Invariant subspace problem      112 123
Invertible linear operator      100
Isolated point in a metric space      16
Jordan, Camille      43
Klein, Felix      27
Kolmogorov, Andrei      33
Laguerre functions      78
Laguerre, Edmond      78
Laurent series      105
Law of Large Numbers      33
Lebesgue integral of a function on M      46
Lebesgue measurable subsets      42
Lebesgue measure      34 42
Lebesgue space $L^{p}(X, \mu)$      59
Lebesgue, Henri      9 32 42
Lebesgue’s dominated convergence, theorem      53 64
Lebesgue’s monotone convergence, theorem      50
Left shift      93 100
Left shift is not compact      110
Left shift, spectrum of      111
Legendre polynomials      78
Legendre, Adrien — Marie      78
Levy, Paul P.      1
Limit point in a metric space      16
Linear functional      94 151 153
Linear operator      92
Linear space      3
Liouville, Joseph      103
Liouville’s theorem      103 104
Lipschitz condition      163
Lipschitz, Rudolf Otto Sigismund      163
Lomonosov, Victor      120
Lomonosov’s theorem      120
Machado, Silvio      145
Maximal ideal      165
Measurable function on R      44
Measurable sets      34
Measure      34 36
Measure space      36
Metric      4
Metric space      4
Minkowski, Hermann      26
Minkowski’s inequality      64
Monge, Gaspard      80
Morgenstem, Oskar      175
Multiplication operator      110
Norm      5
Norm of a bounded operator      96
Normalized vector in an inner product, space      76
Normed algebra      99
Normed linear space      5
Nowhere dense set      147
Observable      101 167
Open ball      14
Open ball in a metric space      16
Open cover of sets      16
Open mapping theorem      151
Open set      16
Orthogonal complement      115
Orthogonal vectors in an inner product, space      76
Orthonormal basis      77
Orthonormal sequence      76
Oscillation of a function      149
Outer measure      37 38
Pairwise disjoint collection of sets      186
Parallelogram equality      7 14
Parseval, Marc — Antoine      83
Parseval’s theorem      83
Partially ordered set      154
Peano, Giuseppe      3
Picard, Emile      160
Picard’s iterative method      161
Plancherel identity      168
Plancherel, Michel      168
Poincare, Henri      160
probability      33 136
Probability measure      58
Problems      26
Projection in an inner product space      90
Projection operator      128
Proper ideal      165
quantum mechanics      101 167
Quasinilpotent operator      112 121 132
R-ball centered at x      16
Ransford, Thomas      145
Read, Charles      112
Real part of a complex number      183
Riemann integral      54
Riemann, Georg Friedrich Bernhard      32
Riesz theory for compact operators      108
Riesz — Fischer theorem      84
Riesz — Frechet theorem      115
Riesz, Frigyes      22 32 59 67 68 107 128
Riesz, Marcel      22
Right shift      93 100
Right shift, invariant subspace of      113
Right shift, spectrum of      111
Ring      165
Ring of sets      35
Ring with unit      165
Rosetta stone      80
Runge, Carle      145
Runge’s theorem      145
Schauder basis      11 123
Schauder fixed point theorem      121
Schrodinger, Erwin      167
Schwarz, Hermann      7
Scottish Cafe      28
Second category set      147
Separable metric space      22
Sequentially compact      16
Shift operator      93
Simple function      45
Spectral radius      105
Spectral radius formula      105 112
Spectral theorem for compact Hermitian operators      127
Spectral theory      101
Spectrum      101
State function      167
Steinhaus, Hugo      28 61
Step function      67
Stone — Weierstrass theorem      140
Stone, Marshall      128 145
Strong Law of Large Numbers      35
Subcover of sets      16
Submultiplicative norm      99
Supremum of a set      9
Symmetric difference of sets      38
Symmetric, A-symmetric subset      144
Time-limited signal      172
Totally bounded set      18
Totally ordered set      155
Translation invariance of Lebesgue, measure      158
Trigonometric polynomials      86
Trigonometric system      76 86 88
Uncertainty principle      169 172
Uncountable set      186
Uniform algebra      145
Uniform boundedness principle      153
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