Linear Operators: General theory |
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Page 426
... unit sphere of X is a metric topology if and only if X * is separable . PROOF . Let X * be separable , and let x : XX ** be the natural embedding . By Theorem 1 , the X * topology of the closed unit sphere S ** of X ** is a metric ...
... unit sphere of X is a metric topology if and only if X * is separable . PROOF . Let X * be separable , and let x : XX ** be the natural embedding . By Theorem 1 , the X * topology of the closed unit sphere S ** of X ** is a metric ...
Page 458
... unit sphere of co contains no extremal points . 4 Let ( S , E , μ ) be a σ - finite positive measure space . Show that L1 ( S , E , u ) is isometrically isomorphic to the conjugate * of a B - space X if and only if S can be written as a ...
... unit sphere of co contains no extremal points . 4 Let ( S , E , μ ) be a σ - finite positive measure space . Show that L1 ( S , E , u ) is isometrically isomorphic to the conjugate * of a B - space X if and only if S can be written as a ...
Page 485
... unit sphere S * of Y * is Y - compact ( V.4.2 ) , it follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in ... unit spheres in X , X ** , respectively , and if x is the natural embedding of X into X ** , then by Theorem V.4.5 ...
... unit sphere S * of Y * is Y - compact ( V.4.2 ) , it follows from Lemma 7 and Lemma I.5.7 that T * S * is compact in ... unit spheres in X , X ** , respectively , and if x is the natural embedding of X into X ** , then by Theorem V.4.5 ...
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