Linear Operators: General theory |
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Page 114
... Statement ( b ) is clear from the definitions . Statement ( c ) is evident for μ - integrable simple functions ; to prove it in the general case , let fn be a sequence of finitely valued μ - simple functions which determine g . Then the ...
... Statement ( b ) is clear from the definitions . Statement ( c ) is evident for μ - integrable simple functions ; to prove it in the general case , let fn be a sequence of finitely valued μ - simple functions which determine g . Then the ...
Page 415
... Statement ( iii ) follows from ( i ) and ( ii ) . We now prove ( iv ) . Statement ( i ) and Lemma 3 show that co ( A ) + co ( B ) is convex and closed , so that co ( A + B ) C co ( 4 ) + co ( B ) . Now , since x + y is a continuous ...
... Statement ( iii ) follows from ( i ) and ( ii ) . We now prove ( iv ) . Statement ( i ) and Lemma 3 show that co ( A ) + co ( B ) is convex and closed , so that co ( A + B ) C co ( 4 ) + co ( B ) . Now , since x + y is a continuous ...
Page 447
... Statement ( b ) follows from the inequality 21 ( x + - a 2 ( Y1 + Y2 ) ) ≤ f ( x + ay1 ) + ( x + ay2 ) . Statement ( c ) is trivial . Statement ( d ) follows from the inequality T ( x , y ) + T ( x , y ) ≥ t ( x , 0 ) = 0 · T ( x , y ) ...
... Statement ( b ) follows from the inequality 21 ( x + - a 2 ( Y1 + Y2 ) ) ≤ f ( x + ay1 ) + ( x + ay2 ) . Statement ( c ) is trivial . Statement ( d ) follows from the inequality T ( x , y ) + T ( x , y ) ≥ t ( x , 0 ) = 0 · T ( x , y ) ...
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