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 limm , 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 space is a Cauchy sequence if limm , 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 ...
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 1.6.10 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 1.6.10 21 METRIC ...
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