Linear Operators: General theory |
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Page 425
... sphere of X ** . It follows immediately that it contains every point in X ** . Q.E.D. Theorems 2 and 5 lead to an important result on reflexive spaces . 7 THEOREM . A B - space is reflexive if and only if its closed unit sphere is ...
... sphere of X ** . It follows immediately that it contains every point in X ** . Q.E.D. Theorems 2 and 5 lead to an important result on reflexive spaces . 7 THEOREM . A B - space is reflexive if and only if its closed unit sphere is ...
Page 458
... sphere of c 。 contains no extremal points . 4 Let ( S , 2 , μ ) be a σ - finite positive measure space . Show that L1 ( S , E , μ ) is isometrically isomorphic to the conjugate * of a B - space if and only if S can be written as a ...
... sphere of c 。 contains no extremal points . 4 Let ( S , 2 , μ ) be a σ - finite positive measure space . Show that L1 ( S , E , μ ) is isometrically isomorphic to the conjugate * of a B - space if and only if S can be written as a ...
Page 485
... 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 the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 ...
... 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 the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 ...
Contents
B Topological Preliminaries | 10 |
Metric Spaces | 23 |
Product Spaces | 31 |
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 compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations Doklady Akad Duke Math E₁ elements ergodic exists extension f₁ function defined function f Hausdorff space Hence Hilbert space homomorphism inequality 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 o-field 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 theory topological space u-integrable u-measurable u-null uniformly unit sphere valued function vector space weakly compact zero ΕΕΣ