Linear Operators: General theory |
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Page 10
... neighborhood of the point p is an open set 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 ...
... neighborhood of the point p is an open set 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 ...
Page 51
... neighborhood of f ( y ) , then , by Lemma 2 ( c ) , Vf ( y - 1x ) is a neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) C Vf ( y - 1x ) , then Ux - 1y is a neighborhood of y such that f ( Ux − 1y ) = f ( U ) f ...
... neighborhood of f ( y ) , then , by Lemma 2 ( c ) , Vf ( y - 1x ) is a neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) C Vf ( y - 1x ) , then Ux - 1y is a neighborhood of y such that f ( Ux − 1y ) = f ( U ) f ...
Page 56
... neighborhood M of 0 such that M - MCG . For every xe X , x / n → 0 , and so xe nM for large n . Thus € 00 X = UnM ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the origin ...
... neighborhood M of 0 such that M - MCG . For every xe X , x / n → 0 , and so xe nM for large n . Thus € 00 X = UnM ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the origin ...
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