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 Q ( 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 Q ( 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
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ