Linear Operators: General theory |
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Page 483
... weakly compact , then TS is compact in the Y * topology of Y and thus x ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) ≤ × ( TS ) . According to Theorem V.4.5 ...
... weakly compact , then TS is compact in the Y * topology of Y and thus x ( TS ) is compact and hence closed in the Y * topology of Y ** . Thus if T is weakly compact , ( i ) yields T ** ( S1 ) ≤ × ( TS ) . According to Theorem V.4.5 ...
Page 494
... weakly compact set of rca ( S ) , and therefore T * is a weakly compact operator . By Theorem 4.8 this implies that T is a weakly compact operator . Q.E.D. 4 THEOREM . If T is a weakly compact operator from C ( S ) to X , then T sends weak ...
... weakly compact set of rca ( S ) , and therefore T * is a weakly compact operator . By Theorem 4.8 this implies that T is a weakly compact operator . Q.E.D. 4 THEOREM . If T is a weakly compact operator from C ( S ) to X , then T sends weak ...
Page 540
... Weakly compact operators from C [ 0 , 1 ] to X were treated by Sirvint [ 3 ] . A very incisive discussion of weakly compact operators with domain C ( S ) was given by Grothendieck [ 4 ] who proved Theorems 7.4-7.6 by other means ...
... Weakly compact operators from C [ 0 , 1 ] to X were treated by Sirvint [ 3 ] . A very incisive discussion of weakly compact operators with domain C ( S ) was given by Grothendieck [ 4 ] who proved Theorems 7.4-7.6 by other means ...
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A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ