Linear Operators: General theory |
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Page 10
... containing p . A neighborhood of the set A is an open set containing A. If A is a subset of X , then a point p is a limit point , or a point of accumulation , of A provided every neighborhood of p contains at least one point qp , with ...
... containing p . A neighborhood of the set A is an open set containing A. If A is a subset of X , then a point p is a limit point , or a point of accumulation , of A provided every neighborhood of p contains at least one point qp , with ...
Page 20
... contains a non - void open set . Thus A1 ‡ X , and A ' is open , and contains a sphere S1 = S ( P1 , ɛ1 ) with 0 < & 1 < 1/2 . By assumption , the set A2 does not contain the open set S ( p1 , & 1 / 2 ) ; hence the non - void open set ...
... contains a non - void open set . Thus A1 ‡ X , and A ' is open , and contains a sphere S1 = S ( P1 , ɛ1 ) with 0 < & 1 < 1/2 . By assumption , the set A2 does not contain the open set S ( p1 , & 1 / 2 ) ; hence the non - void open set ...
Page 309
... contains no atoms having measure greater than ɛ . It will now be shown that every measurable subset B of A contains a set F with 0 < u ( F ) ≤ ε . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive ...
... contains no atoms having measure greater than ɛ . It will now be shown that every measurable subset B of A contains a set F with 0 < u ( F ) ≤ ε . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive ...
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