Linear Operators: General theory |
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Page 61
... seen that Ta exists for each x , and that T is continuous . Thus Lemma 4 shows that T is bounded . Now , when Tx exists , and thus Tx = lim | T | ≤lim inf | T || x | , 8 个袋 T≤lim inf | T | · Finally , to verify that sup , | T ...
... seen that Ta exists for each x , and that T is continuous . Thus Lemma 4 shows that T is bounded . Now , when Tx exists , and thus Tx = lim | T | ≤lim inf | T || x | , 8 个袋 T≤lim inf | T | · Finally , to verify that sup , | T ...
Page 254
... seen that vg is equivalent to vg and thus that v is in V. Since { v } is a basis , the vector u has an expansion of the form ua Σβικα , ( u , v ) , so that u , is in the closed linear manifold de- termined by those v with ( u , v ) 0 ...
... seen that vg is equivalent to vg and thus that v is in V. Since { v } is a basis , the vector u has an expansion of the form ua Σβικα , ( u , v ) , so that u , is in the closed linear manifold de- termined by those v with ( u , v ) 0 ...
Page 449
... seen that ( aA1 + ( 1 — a ) A ) ○ B1 = $ for 0 < a < 1 . Since a41 + ( 1 - a ) A is open , ( aД1 + ( 1 — a ) A ) ○ B = 4 for 0 < a < 1. Let p e A1 and q € A ; since p is interior to A , ( 1 - a ) p + aq is in A for all sufficiently ...
... seen that ( aA1 + ( 1 — a ) A ) ○ B1 = $ for 0 < a < 1 . Since a41 + ( 1 - a ) A is open , ( aД1 + ( 1 — a ) A ) ○ B = 4 for 0 < a < 1. Let p e A1 and q € A ; since p is interior to A , ( 1 - a ) p + aq is in A for all sufficiently ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ