Linear Operators: General theory |
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Page 245
... finite dimensional normed linear space is continuous . ... 9 PROOF . Let { b1 , b } be a Hamel basis for the finite dimension- al normed linear space X so that every x in X has a unique represen- tation in the form a a1b1 + ... + anbn ...
... finite dimensional normed linear space is continuous . ... 9 PROOF . Let { b1 , b } be a Hamel basis for the finite dimension- al normed linear space X so that every x in X has a unique represen- tation in the form a a1b1 + ... + anbn ...
Page 246
... finite . Then the dimension of X ** is finite , and , since X is equivalent to a subspace of X ** ( II.3.19 ) , the dimension of X is finite . Hence , from the first part of this proof , X and X * have the same dimension . Q.E.D. In the ...
... finite . Then the dimension of X ** is finite , and , since X is equivalent to a subspace of X ** ( II.3.19 ) , the dimension of X is finite . Hence , from the first part of this proof , X and X * have the same dimension . Q.E.D. In the ...
Page 849
... finite , III.4.3 ( 126 ) Lebesgue extension of , III.5.18 ( 143 ) as a metric space , III.7.1 ( 158 ) , III.9.6 ( 169 ) positive , III.4.3 ( 126 ) product , of finite number of finite measure spaces , III.11.3 ( 186 ) of finite number of o ...
... finite , III.4.3 ( 126 ) Lebesgue extension of , III.5.18 ( 143 ) as a metric space , III.7.1 ( 158 ) , III.9.6 ( 169 ) positive , III.4.3 ( 126 ) product , of finite number of finite measure spaces , III.11.3 ( 186 ) of finite number of o ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ