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 said to be complete . The next three lemmas are immediate consequences of the definitions . 6 LEMMA . In a metric space , a convergent sequence is a Cauchy sequence . A Cauchy sequence converges if and only if it has a ...
... metric space is said to be complete . The next three lemmas are immediate consequences of the definitions . 6 LEMMA . In a metric space , a convergent sequence is a Cauchy sequence . A Cauchy sequence converges if and only if it has a ...
Page 91
... metric . Thus every complete linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in ...
... metric . Thus every complete linear metric space can be metrized to be an F - space . Further , a normed linear space is a B - space provided it is complete under some equivalent metric . See also van Dantzig [ 1 ] , [ 2 ] . Norms in ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ