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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ