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 α , 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 α , 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 I topology . Note that 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 I topology . Note that may already be a linear ...
Contents
Exercises | 9 |
Definitions and Basic Properties | 13 |
Metric Spaces | 19 |
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 closed sets compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense disjoint Doklady Akad E₁ elements ergodic exists f₁ finite number follows function defined function f Hausdorff space Hence Hilbert space homomorphism 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 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 Theorem theory topological space u-integrable u-measurable u-null uniformly unit sphere vector space weakly compact zero ΕΕΣ