Linear Operators: General theory |
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Page 449
... non - zero tangent functional at each point of a dense subset of its boundary . Then A is convex . = PROOF . Denoting the interior of A by A1 , we will show first that there is no non - zero tangent to A at any point of aÃ1 + ( 1 — a ) ...
... non - zero tangent functional at each point of a dense subset of its boundary . Then A is convex . = PROOF . Denoting the interior of A by A1 , we will show first that there is no non - zero tangent to A at any point of aÃ1 + ( 1 — a ) ...
Page 557
... non - zero polynomials S ,, i 2 , . . . , k such that S , ( T ) ; = 0 . If R S1 S2S , then R ( T ) , = 0 , and 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 ...
... non - zero polynomials S ,, i 2 , . . . , k such that S , ( T ) ; = 0 . If R S1 S2S , then R ( T ) , = 0 , and 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 ...
Page 579
... non - zero number in o ( T ) is a pole of T and has finite positive index . For such a number 2 , the projection E ( 2 ) has a non - zero finite dimensional range given by the formula E ( 2 ) X = { x | ( T — ¿ I ) x = 0 } where v is the ...
... non - zero number in o ( T ) is a pole of T and has finite positive index . For such a number 2 , the projection E ( 2 ) has a non - zero finite dimensional range given by the formula E ( 2 ) X = { x | ( T — ¿ I ) x = 0 } where v is the ...
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