Linear Operators: General theory |
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Page 29
... Let D be a directed set , let X and Y be topological spaces , and let Y be complete and metric . Let f D x X → Y , that f ( d , x ) is a generalized sequence of functions on X , with values in Y Suppose that : ( a ) for each do e D , the ...
... Let D be a directed set , let X and Y be topological spaces , and let Y be complete and metric . Let f D x X → Y , that f ( d , x ) is a generalized sequence of functions on X , with values in Y Suppose that : ( a ) for each do e D , the ...
Page 173
... f cannot be represented as f ( s ) = g ( t ) dt with g in L1 ( S , Σ , μ ) . 40 Under the hypotheses of Exercise 36 , let ... Let 1 ≤ p < ∞o , and let ( S , E , μ ) be a measure space . Let fe ƒ ‹ L2 ( S , Σ , μ ) , g € L ( S , Σ , μ ) , 1 ...
... f cannot be represented as f ( s ) = g ( t ) dt with g in L1 ( S , Σ , μ ) . 40 Under the hypotheses of Exercise 36 , let ... Let 1 ≤ p < ∞o , and let ( S , E , μ ) be a measure space . Let fe ƒ ‹ L2 ( S , Σ , μ ) , g € L ( S , Σ , μ ) , 1 ...
Page 211
... { F } . Let F1 be chosen arbitra- rily and suppose F1 , . . . , F already chosen . If ACFU ... UFk Εκ lub 8 ( F ) , where F varies k the lemma is satisfied . Otherwise , let over all sets F in F satisfying - S ( F , 8 ( F ) ) F ; = 4 , k ...
... { F } . Let F1 be chosen arbitra- rily and suppose F1 , . . . , F already chosen . If ACFU ... UFk Εκ lub 8 ( F ) , where F varies k the lemma is satisfied . Otherwise , let over all sets F in F satisfying - S ( F , 8 ( F ) ) F ; = 4 , k ...
Contents
Preliminary Concepts | 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 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 ergodic exists f₁ finite dimensional function defined function f 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-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ