Linear Operators: General theory |
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Page 151
... Consequently , lim sup fn - fm , ≤ lim sup { \ † n ( s ) — † m ( s ) Pv ( u , ds ) } 1 + 2 × 1o m , n∞ m , n∞ E8 1 / p lim sup [ { \ † n ( 8 ) — † ( 3 ) \ ” v ( μ , ds ) } 1o m , n∞ + { { £ ̧ \ fm ( 8 ) − † ( s ) \ P v ( μ , ds ) ) ...
... Consequently , lim sup fn - fm , ≤ lim sup { \ † n ( s ) — † m ( s ) Pv ( u , ds ) } 1 + 2 × 1o m , n∞ m , n∞ E8 1 / p lim sup [ { \ † n ( 8 ) — † ( 3 ) \ ” v ( μ , ds ) } 1o m , n∞ + { { £ ̧ \ fm ( 8 ) − † ( s ) \ P v ( μ , ds ) ) ...
Page 254
... Consequently { u } and { v } have the same cardinality . Q.E.D. 15 DEFINITION . The cardinality of an arbitrary orthonormal basis of a Hilbert space is its dimension . 16 THEOREM . Two Hilbert spaces are isometrically isomorphic if and ...
... Consequently { u } and { v } have the same cardinality . Q.E.D. 15 DEFINITION . The cardinality of an arbitrary orthonormal basis of a Hilbert space is its dimension . 16 THEOREM . Two Hilbert spaces are isometrically isomorphic if and ...
Page 637
... Consequently , ( i ) ,. . . , ( v ) are proved inductively for all n . We now obtain an estimate for the series ox ( n ) ( t ) which majorizes o Sn ( t ) . By Lemma 20 ( c ) , for each w > wo there exists a constant M∞ such that y ( t ) ...
... Consequently , ( i ) ,. . . , ( v ) are proved inductively for all n . We now obtain an estimate for the series ox ( n ) ( t ) which majorizes o Sn ( t ) . By Lemma 20 ( c ) , for each w > wo there exists a constant M∞ such that y ( t ) ...
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