Linear Operators: General theory |
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Page 253
... correspond- ing equivalence classes and that u U. Consider an arbitrary ele- ment υβ 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 B α # υβ u , Ug in U ...
... correspond- ing equivalence classes and that u U. Consider an arbitrary ele- ment υβ 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 B α # υβ u , Ug 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 U1 = { 8 || gk ̧ ( 8 ) —fo ( s ) | > εo } is an open set containing so for i = 1 , . . . , r . Since A is dense in S , there exists a point s ...
... corresponding to ε and k = 1 guaranteed by the quasi - uniform convergence of { g } . Then U1 = { 8 || gk ̧ ( 8 ) —fo ( s ) | > εo } is an open set containing so for i = 1 , . . . , r . Since A is dense in S , there exists a point s ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz Russian S₁ scalar semi-group sequentially compact Show subset subspace Suppose Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ