Linear Operators: General theory |
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Page 51
... neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) Vf ( y - 1x ) , then Ux - 1y is a neighborhood of y such that f ( Ux ̄1y ) = f ( U ) f ( x − 1y ) C Vf ( y ̄1x ) f ( x - 1y ) = V. Therefore , ƒ is con- f tinuous ...
... neighborhood of f ( x ) . If U is a neighborhood of a such that f ( U ) Vf ( y - 1x ) , then Ux - 1y is a neighborhood of y such that f ( Ux ̄1y ) = f ( U ) f ( x − 1y ) C Vf ( y ̄1x ) f ( x - 1y ) = V. Therefore , ƒ is con- f tinuous ...
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 ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite 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 theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ