Linear Operators: General theory |
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Page 464
... satisfies inf Rf ( x ) RP ( f ) ≤ sup Rf ( x ) , x € K also satisfies for some xo € € K. P ( f ) = f ( xo ) , xЄK fer fer ( 4 ) If K¿ , § ≤ 0 , 0 a limit ordinal , is a monotone decreasing trans- finite sequence of T - closed convex ...
... satisfies inf Rf ( x ) RP ( f ) ≤ sup Rf ( x ) , x € K also satisfies for some xo € € K. P ( f ) = f ( xo ) , xЄK fer fer ( 4 ) If K¿ , § ≤ 0 , 0 a limit ordinal , is a monotone decreasing trans- finite sequence of T - closed convex ...
Page 498
... satisfies ( i ) then ( ii ) defines an operator Ton X to L ( S , E , μ ) whose norm satisfies ( iii ) . Furthermore T is weakly compact if and only if x * ( · ) is countably additive on in the strong topology of X * . PROOF . If , for E ...
... satisfies ( i ) then ( ii ) defines an operator Ton X to L ( S , E , μ ) whose norm satisfies ( iii ) . Furthermore T is weakly compact if and only if x * ( · ) is countably additive on in the strong topology of X * . PROOF . If , for E ...
Page 557
... satisfies R1 ( T ) = 0. In the same way , the product R2 of all the factors ( 2-2 ) , where Bi min ( ¤¿ , v ( ^ ¿ ) ) , satisfies R2 ( T ) = 0. Since any polynomial P having a zero of order v ( 2 ) at each λ € σ ( T ) is divisible by R2 ...
... satisfies R1 ( T ) = 0. In the same way , the product R2 of all the factors ( 2-2 ) , where Bi min ( ¤¿ , v ( ^ ¿ ) ) , satisfies R2 ( T ) = 0. Since any polynomial P having a zero of order v ( 2 ) at each λ € σ ( T ) is divisible by R2 ...
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