Linear Operators: General theory |
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Page 40
... unit e , then an element a 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 2 , is actually ...
... unit e , then an element a 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 2 , is actually ...
Page 41
... unit , then R / M is isomorphic with the field P2 . 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 P2 . 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 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 the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma ...
... 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 the X ** topology of X * . Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra analytic arbitrary B-space B₁ ba(S Banach Borel sets ca(S Cauchy sequence closed unit sphere compact Hausdorff space compact operator complex numbers conditionally compact contains continuous functions convex set Corollary countably additive DEFINITION denote dense E₁ element equation equivalent exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue measure Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space neighborhood non-negative non-zero normed linear space o-field o-finite open set operator topology positive measure space properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly valued function weak topology weakly compact weakly sequentially compact X₁ zero ΕΕΣ