Linear Operators: General theory |
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Page 19
... metric topology in X is the smallest topology contain- ing the spheres . The set X , with its metric topology , is called a metric space . If X is a topological space such that there exists a metric func- tion whose topology is the same ...
... metric topology in X is the smallest topology contain- ing the spheres . The set X , with its metric topology , is called a metric space . If X is a topological space such that there exists a metric func- tion whose topology is the same ...
Page 20
... metric space is a Cauchy sequence if lim , n ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The next three lemmas are immediate consequences of the definitions . 6 LEMMA . In a metric ...
... metric space is a Cauchy sequence if lim , n ( am , an ) = 0. If every Cauchy se- quence is convergent , a metric space is said to be complete . The next three lemmas are immediate consequences of the definitions . 6 LEMMA . In a metric ...
Page 21
... metric space X. By Lemma 5.7 ( c ) , A is closed . If A is not sequentially compact , then some sequence { a } contains no convergent subsequence . Hence each point in A has a neighborhood containing at most a finite I.6.10 21 METRIC ...
... metric space X. By Lemma 5.7 ( c ) , A is closed . If A is not sequentially compact , then some sequence { a } contains no convergent subsequence . Hence each point in A has a neighborhood containing at most a finite I.6.10 21 METRIC ...
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