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 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 u - 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 ( 4 ) + co ( B ) is convex and closed , so that co ( A + B ) Cco ( 4 ) + co ( B ) . Now , since a + 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 ) Cco ( 4 ) + co ( B ) . Now , since a + y is a continuous ...
Page 487
... statement is , in general , false . This will be seen in an exercise . However the dual statement is true if the range UX is closed , in which case the range U ** is also closed . Dually , if U * Y * is closed , so is UX . These results ...
... statement is , in general , false . This will be seen in an exercise . However the dual statement is true if the range UX is closed , in which case the range U ** is also closed . Dually , if U * Y * is closed , so is UX . These results ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz 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