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 f , be a sequence of finitely valued μ - simple functions which determine g . Then ...
... Statement ( b ) is clear from the definitions . Statement ( c ) is evident for u - integrable simple functions ; to prove it in the general case , let f , be a sequence of finitely valued μ - simple functions which determine g . Then ...
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
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