Linear Operators: General theory |
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Page 29
... Let D be a directed set , let X and Y be topological spaces , and let Y be complete and metric . Let f D x XY , so that f ( d , x ) is a generalized sequence of functions on X , with values in Y Suppose that : ( a ) for each do e D , the ...
... Let D be a directed set , let X and Y be topological spaces , and let Y be complete and metric . Let f D x XY , so that f ( d , x ) is a generalized sequence of functions on X , with values in Y Suppose that : ( a ) for each do e D , the ...
Page 211
... { F } . Let F1 be chosen arbitra- rily and suppose F1 , ... , F already chosen . If ACFU ... UFx the lemma is satisfied . Otherwise , let & lub 8 ( F ) , where F varies over all sets F in F satisfying = S ( F , 8 ( F ) ) F ; = & , φ , k U ...
... { F } . Let F1 be chosen arbitra- rily and suppose F1 , ... , F already chosen . If ACFU ... UFx the lemma is satisfied . Otherwise , let & lub 8 ( F ) , where F varies over all sets F in F satisfying = S ( F , 8 ( F ) ) F ; = & , φ , k U ...
Page 223
... Let ƒ be a continuous increasing function on an open interval ( c , d ) , with f ( c ) = a , f ( d ) = b . Show that the Lebesgue - Stieltjes integral exists and is equal to I. d S " ( f ( x ) ) dh ( f ( x ) ) g 6 Show that a monotone ...
... Let ƒ be a continuous increasing function on an open interval ( c , d ) , with f ( c ) = a , f ( d ) = b . Show that the Lebesgue - Stieltjes integral exists and is equal to I. d S " ( f ( x ) ) dh ( f ( x ) ) g 6 Show that a monotone ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
Copyright | |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ