Linear Operators: General theory |
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Page 14
... topological spaces , and if f : X → Y and g : YZ are continuous , then the ... space X , and let a be a scalar . Then the functions given by the ... topological space , a continuous real function which is not a constant ? If x and y are ...
... topological spaces , and if f : X → Y and g : YZ are continuous , then the ... space X , and let a be a scalar . Then the functions given by the ... topological space , a continuous real function which is not a constant ? If x and y are ...
Page 51
... topological space X is bounded if , given any neighborhood V of the zero in X , there exists a positive real number & such that aB CV providing aε . 8 LEMMA . A compact subset of a linear topological space is bounded . PROOF . Let B be ...
... topological space X is bounded if , given any neighborhood V of the zero in X , there exists a positive real number & such that aB CV providing aε . 8 LEMMA . A compact subset of a linear topological space is bounded . PROOF . Let B be ...
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 ...
... 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 ...
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