Linear Operators: General theory |
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Page 143
... Borel measure in [ a , b ] . The Lebesgue measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure ...
... Borel measure in [ a , b ] . The Lebesgue measurable sets in [ a , b ] are the sets in the Lebesgue extension ( relative to Borel measure ) of the o - field of Borel sets in [ a , b ] . Similarly , the Lebesgue - Stieltjes measure ...
Page 492
... Borel sets in E and all finite sets of scalars a1 , . . . , an with a , | ≤ 1 . We are now prepared to represent the general operator . 2 THEOREM . Let S be a compact Hausdorff space and let T be an operator on C ( S ) to X. Then there ...
... Borel sets in E and all finite sets of scalars a1 , . . . , an with a , | ≤ 1 . We are now prepared to represent the general operator . 2 THEOREM . Let S be a compact Hausdorff space and let T be an operator on C ( S ) to X. Then there ...
Page 516
... Borel sets of S which is not identically zero and is 4 - invariant for every фє G. 42 Show that in Exercise 38 the set function u is unique up to a positive constant factor if and only if n - 1 ( p ( s ) ) converges uniformly to a ...
... Borel sets of S which is not identically zero and is 4 - invariant for every фє G. 42 Show that in Exercise 38 the set function u is unique up to a positive constant factor if and only if n - 1 ( p ( s ) ) converges uniformly to a ...
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