Linear Operators: General theory |
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Page 415
... Statement ( iii ) follows from ( i ) and ( ii ) . We now prove ( iv ) . Statement ( i ) and Lemma 3 show that co ( A ) + co ( B ) is convex and closed , so that co ( A + B ) C 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 ( A ) + co ( B ) is convex and closed , so that co ( A + B ) C 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 ) +7 ( x , y ) ≥ τ ( 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 ) +7 ( x , y ) ≥ τ ( 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 ...
Page 487
... statement that T ( S ) is an equicontinuous subset of C ( S * ) . It follows from Theorem IV.6.7 , that T ( S ) is ... statement is , in general , false . This will be seen in an exercise . However the dual statement is true if the range ...
... statement that T ( S ) is an equicontinuous subset of C ( S * ) . It follows from Theorem IV.6.7 , that T ( S ) is ... statement is , in general , false . This will be seen in an exercise . However the dual statement is true if the range ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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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 Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ