Linear Operators: General theory |
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Page 58
... linear , and continuous ( I.8.3 ) . Hence , by Theorem 2 , its inverse pr1 is continuous . Thus T = propr1 is continuous ( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space under each of two metrics , and if one of the ...
... linear , and continuous ( I.8.3 ) . Hence , by Theorem 2 , its inverse pr1 is continuous . Thus T = propr1 is continuous ( I.4.17 ) . Q.E.D. 5 THEOREM . If a linear space is an F - space under each of two metrics , and if one of the ...
Page 417
... linear functional on a linear topological space separates two sets , one of which has an interior point , then the functional is continuous . = PROOF . Let X be a linear topological space , and 41 , 42 C. Let h be a linear functional ...
... linear functional on a linear topological space separates two sets , one of which has an interior point , then the functional is continuous . = PROOF . Let X be a linear topological space , and 41 , 42 C. Let h be a linear functional ...
Page 418
... continuous linear functional separates K and p . 13 COROLLARY . If p and q are distinct points of a locally convex linear topological space X , there is a continuous linear functional f de- fined on such that f ( p ) #f ( q ) . As a ...
... continuous linear functional separates K and p . 13 COROLLARY . If p and q are distinct points of a locally convex linear topological space X , there is a continuous linear functional f de- fined on such that f ( p ) #f ( q ) . As a ...
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