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 , 1 ) with 0 < & < 1/2 . By assumption , the set A does not contain the open set S ( p1 , & 1 / 2 ) ; hence the non - void open set AS ( P1 ...
... contains a non - void open set . Thus A1 X , and A ' is open , and contains a sphere S1 S ( P1 , 1 ) with 0 < & < 1/2 . By assumption , the set A does not contain the open set S ( p1 , & 1 / 2 ) ; hence the non - void open set AS ( P1 ...
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 < μ ( 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 < μ ( F ) ≤ ε . Sup- pose , on the contrary , that some such set B contains no set of Σ with positive ...
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