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
A Settheoretic Preliminaries | 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 Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ