Linear Operators: General theory |
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Page 114
... Statement ( b ) is clear from the definitions . Statement ( c ) is evident for u - integrable simple functions ; to prove it in the general case , let f2 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 u - integrable simple functions ; to prove it in the general case , let f2 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 ( 4 ) + co ( B ) is convex and closed , so that co ( A + B ) ≤ 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 ( 4 ) + co ( B ) is convex and closed , so that co ( A + B ) ≤ co ( 4 ) + co ( B ) . Now , since x + y is a continuous ...
Page 447
... Statement ( c ) is trivial . Statement ( d ) follows from the inequality T ( x , y ) + T ( x , y ) ≥ t ( x , 0 ) = 0 ⋅ t ( x , y ) = 0 . Statement ( e ) is trivial . Q.E.D. = 4 DEFINITION . If A is a subset of a linear space X , and x ...
... Statement ( c ) is trivial . Statement ( d ) follows from the inequality T ( x , y ) + T ( x , y ) ≥ t ( x , 0 ) = 0 ⋅ t ( x , y ) = 0 . Statement ( e ) is trivial . Q.E.D. = 4 DEFINITION . If A is a subset of a linear space X , and x ...
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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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