Linear Operators: General theory |
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Page 151
... Consequently , lim sup \ fn - fm \ p ≤ lim sup { \ fn ( s ) — † m ( s ) 3 v ( μ , ds ) } 1 + 2ɛ1o m , n∞ lim sup m.n∞0 Εε [ { ƒ £ ̧ \ † n ( $ ) — † ( $ ) \ Pv ( μ , ds ) } 1P + { { £ ̧ \ fm ( 8 ) − † ( 8 ) | P v ( μ , ds ) } 13 ] + 2 × ...
... Consequently , lim sup \ fn - fm \ p ≤ lim sup { \ fn ( s ) — † m ( s ) 3 v ( μ , ds ) } 1 + 2ɛ1o m , n∞ lim sup m.n∞0 Εε [ { ƒ £ ̧ \ † n ( $ ) — † ( $ ) \ Pv ( μ , ds ) } 1P + { { £ ̧ \ fm ( 8 ) − † ( 8 ) | P v ( μ , ds ) } 13 ] + 2 × ...
Page 557
... consequently R ( T ) x = 0 for all x X. Thus a non - zero polynomial R exists such that R ( T ) = • = 0 . Let R be factored as R ( 2 ) = ẞII - 1 ( λ —λ ̧ ) a2 . If  ̧ ¢ œ ( T ) , then ( T — ¿ ¡ I ) x = 0 implies x = 0. Consequently ...
... consequently R ( T ) x = 0 for all x X. Thus a non - zero polynomial R exists such that R ( T ) = • = 0 . Let R be factored as R ( 2 ) = ẞII - 1 ( λ —λ ̧ ) a2 . If  ̧ ¢ œ ( T ) , then ( T — ¿ ¡ I ) x = 0 implies x = 0. Consequently ...
Page 615
... consequently m - · n U ( t ) = mU ( t / n ) = U ( mt / n ) for each rational number m / n with 0 ≤m / n≤ 1 , and each t in the interval 0 ≤t ≤ ɛ . In particular , m / n U ( e ) = U ( ɛm / n ) , so that , by continuity , tU ( ε ) = U ...
... consequently m - · n U ( t ) = mU ( t / n ) = U ( mt / n ) for each rational number m / n with 0 ≤m / n≤ 1 , and each t in the interval 0 ≤t ≤ ɛ . In particular , m / n U ( e ) = U ( ɛm / n ) , so that , by continuity , tU ( ε ) = U ...
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