Linear Operators: General theory |
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Page 51
... linear topological space . Then the denumerable set of vectors of the form ax + ... + ẞy with a , Po and x ... linear topological space X is bounded if , given any neighborhood V of the zero in X , there exists a positive real number ...
... linear topological space . Then the denumerable set of vectors of the form ax + ... + ẞy with a , Po and x ... linear topological space X is bounded if , given any neighborhood V of the zero in X , there exists a positive real number ...
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 X. 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 X. Let h be a linear functional ...
Page 419
... topology of X. The following lemma is a consequence of Definition 2 . 3 LEMMA . If I is a total linear space of linear functionals on X , X is a locally convex linear topological space in its П topology . Note that X may already be a linear ...
... topology of X. The following lemma is a consequence of Definition 2 . 3 LEMMA . If I is a total linear space of linear functionals on X , X is a locally convex linear topological space in its П topology . Note that X may already be a linear ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ