Linear Operators: General theory |
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Page 245
... finite dimensional normed linear space is continuous . ... 9 ท PROOF . Let ( b1 , · b1 } 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 x = a1b1 + ... + ...
... finite dimensional normed linear space is continuous . ... 9 ท PROOF . Let ( b1 , · b1 } 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 x = a1b1 + ... + ...
Page 290
... finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( En ) = L1 ( En , Σ ( En ) , μ ) , we obtain a sequence { g } of functions in L such that gno≤ x * , gn ( 8 ) ...
... finite , and let E , be an increasing sequence of measurable sets of finite measure whose union is S. Using the theorem for L1 ( En ) = L1 ( En , Σ ( En ) , μ ) , we obtain a sequence { g } of functions in L such that gno≤ x * , gn ( 8 ) ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f g₁ Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ