Linear Operators: General theory |
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Page 424
... unit sphere in the conjugate space X * of the B - space X is compact in the X topology of X * . PROOF . By Definition II.3.5 , the unit sphere in X * is the set { ƒ ƒ € X * , f ( x ) ≤ x } , and thus the theorem follows from Lemma 1 ...
... unit sphere in the conjugate space X * of the B - space X is compact in the X topology of X * . PROOF . By Definition II.3.5 , the unit sphere in X * is the set { ƒ ƒ € X * , f ( x ) ≤ x } , and thus the theorem follows from Lemma 1 ...
Page 425
... unit 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 ...
... unit 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 512
... unit sphere of B ( X , Y ) is compact in the weak operator topology . Conversely , if the closed unit . sphere of B ( X , Y ) is compact in the weak operator topology , Y is reflexive . 7 Define the BWO topology for B ( X , Y ) to be ...
... unit sphere of B ( X , Y ) is compact in the weak operator topology . Conversely , if the closed unit . sphere of B ( X , Y ) is compact in the weak operator topology , Y is reflexive . 7 Define the BWO topology for B ( X , Y ) to be ...
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