Linear Operators: General theory |
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Page 412
... linear functional on the vector space X , and let be a subspace of X. If f ( 9 ) is not the whole field of scalars , then f ( Y ) = 0 . PROOF . Suppose that there exists a ye with f ( y ) 0. Then f ( xy / f ( y ) ) = x , so that every ...
... linear functional on the vector space X , and let be a subspace of X. If f ( 9 ) is not the whole field of scalars , then f ( Y ) = 0 . PROOF . Suppose that there exists a ye with f ( y ) 0. Then f ( xy / f ( y ) ) = x , so that every ...
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
... 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 last corollary ...
... 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 last corollary ...
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