Linear Operators: General theory |
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Page 40
... unit e , then an element æ in R is called ( right , left ) regular in R in case R contains a ( right , left ) ... unit consists of the integers modulo 2 , i.e. , the two numbers 0 , 1. This Boolean ring , which we denote by P , is actually ...
... unit e , then an element æ in R is called ( right , left ) regular in R in case R contains a ( right , left ) ... unit consists of the integers modulo 2 , i.e. , the two numbers 0 , 1. This Boolean ring , which we denote by P , is actually ...
Page 41
... unit , then R / M is isomorphic with the field 2 . An important example of a Boolean ring with a unit is the ring of subsets of a fixed set . More precisely , let S be a set and let multiplica- tion and addition of arbitrary subsets E ...
... unit , then R / M is isomorphic with the field 2 . An important example of a Boolean ring with a unit is the ring of subsets of a fixed set . More precisely , let S be a set and let multiplica- tion and addition of arbitrary subsets E ...
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 ...
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