Linear Operators: General theory |
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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 - ly 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 - ly is a neighborhood of y such that f ( Ux − 1y ) = f ( U ) f ...
Page 56
... neighborhood M of 0 such that M - M ≤ G. For every xe X , x / n → 0 , and so xe nM for large n . Thus ∞ 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 - M ≤ G. For every xe X , x / n → 0 , and so xe nM for large n . Thus ∞ X = UnM ... neighborhood of 0. Thus the closure of the image of a neighborhood of the origin contains a neighborhood of the origin ...
Page 572
... neighborhood of 21. Then , since ƒ ( σ ( T ) ) = 0 , ƒ has a zero of finite order n at 2. Conse- quently , the function g1 , defined by g1 ( § ) = ( 21 — § ) " / ƒ ( § ) , is analytic in a neighborhood of 21 . Let e be a function ...
... neighborhood of 21. Then , since ƒ ( σ ( T ) ) = 0 , ƒ has a zero of finite order n at 2. Conse- quently , the function g1 , defined by g1 ( § ) = ( 21 — § ) " / ƒ ( § ) , is analytic in a neighborhood of 21 . Let e be a function ...
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