Linear Operators: General theory |
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Page 253
... correspond- ing equivalence classes and that u € U. Consider an arbitrary ele- ment v , in the basis { v } for which ( u , v ) 0. It will be shown that Uge V. Since U and V are corresponding classes there are elements υβ α u , Ug in U ...
... correspond- ing equivalence classes and that u € U. Consider an arbitrary ele- ment v , in the basis { v } for which ( u , v ) 0. It will be shown that Uge V. Since U and V are corresponding classes there are elements υβ α u , Ug in U ...
Page 281
... corresponding to ε and k = 1 guaranteed by the quasi - uniform convergence of { g } . Then U1 = { $ || gx ̧ ( s ) —fo ( $ ) | > εo } = i - " " r . Since A is dense in S , is an open set containing so for i 1 , there exists a point se AU ...
... corresponding to ε and k = 1 guaranteed by the quasi - uniform convergence of { g } . Then U1 = { $ || gx ̧ ( s ) —fo ( $ ) | > εo } = i - " " r . Since A is dense in S , is an open set containing so for i 1 , there exists a point se AU ...
Page 720
... corresponding results for a strongly measurable n- parameter semi - group of operators . 16 Show that the convergence almost everywhere in Theorem 6.9 remains valid if ( S , E , μ ) is a finite measure space and Ss f ( s ) | ( 1+ log + ...
... corresponding results for a strongly measurable n- parameter semi - group of operators . 16 Show that the convergence almost everywhere in Theorem 6.9 remains valid if ( S , E , μ ) is a finite measure space and Ss f ( s ) | ( 1+ log + ...
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