Linear Operators: General theory |
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Page 138
... Consequently , Σ v ( u , E ; ) < ∞ , and i = 1 ∞ ∞ v ( μ , ○ E ; ) = Σv ( μ , E¿ ) → 0 , i = n n - 1 Consequently , — 2 = 1 i = n 00 as n → ∞ . 00 | μ ( E ) — Σ μ ( E ; ) ] = \ μ ( ○ E ; ) | ≤ v ( μ , ○ E ; ) → 0 , အ i = n ...
... Consequently , Σ v ( u , E ; ) < ∞ , and i = 1 ∞ ∞ v ( μ , ○ E ; ) = Σv ( μ , E¿ ) → 0 , i = n n - 1 Consequently , — 2 = 1 i = n 00 as n → ∞ . 00 | μ ( E ) — Σ μ ( E ; ) ] = \ μ ( ○ E ; ) | ≤ v ( μ , ○ E ; ) → 0 , အ i = n ...
Page 151
... Consequently , -- lim sup \ † ‚ — † m ≤ lim sup { m , n∞ m , n∞ Es 1 / p \ fn ( s ) — † m ( s ) Pv ( u , ds ) } 13 + 2ɛ1 » lim sup [ { \ † n ( s ) — † ( 8 ) | Þv ( μ , ds ) } 11o m , n∞ Eε + { ƒ £ ̧ \ † m ( 8 ) − † ( 8 ) \ | P v ( μ ...
... Consequently , -- lim sup \ † ‚ — † m ≤ lim sup { m , n∞ m , n∞ Es 1 / p \ fn ( s ) — † m ( s ) Pv ( u , ds ) } 13 + 2ɛ1 » lim sup [ { \ † n ( s ) — † ( 8 ) | Þv ( μ , ds ) } 11o m , n∞ Eε + { ƒ £ ̧ \ † m ( 8 ) − † ( 8 ) \ | P v ( μ ...
Page 309
... consequently lim ( Fn ) ẞ ( F 。) = = 0. Consequently Thus , the above inequality shows that μ ( F 。) = 0. Let r be an integer such that - F1 , ... , Em + r Fr , and Em + r + 1 = + 1μ ( F ; ) < ɛ and let Urti Fi UF 。. The sets i = r + ...
... consequently lim ( Fn ) ẞ ( F 。) = = 0. Consequently Thus , the above inequality shows that μ ( F 。) = 0. Let r be an integer such that - F1 , ... , Em + r Fr , and Em + r + 1 = + 1μ ( F ; ) < ɛ and let Urti Fi UF 。. The sets i = r + ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ