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 Ve V. Since U and V are corresponding classes there are elements α α υβ υβ U , V 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 Ve V. Since U and V are corresponding classes there are elements α α υβ υβ U , V in U ...
Page 254
... corresponding equivalence classes U and V determine the same closed linear manifold M. Hence , if one of U and V is finite , M is finite dimensional , and therefore the other of U and V is finite and has the same cardinality . If ...
... corresponding equivalence classes U and V determine the same closed linear manifold M. Hence , if one of U and V is finite , M is finite dimensional , and therefore the other of U and V is finite and has the same cardinality . If ...
Page 281
... corresponding to ε and k = 1 guaranteed by the quasi - uniform convergence of { g } . Then { s || gk ̧ ( s ) —fo ( $ ) | > εo } U1 = i is an open set containing so for i = 1 , ... , r . Since A is dense in S , there exists a point se AU ...
... corresponding to ε and k = 1 guaranteed by the quasi - uniform convergence of { g } . Then { s || gk ̧ ( s ) —fo ( $ ) | > εo } U1 = i is an open set containing so for i = 1 , ... , r . Since A is dense in S , there exists a point se AU ...
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