Linear Operators: General theory |
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Page 34
... element ab is called the product of a and b . The product ab is required to satisfy the following conditions : ( i ) a ( bc ) = ( ab ) c , a , b , c e G ; ( ii ) there is an element e in G , called the identity or the unit of G , such ...
... element ab is called the product of a and b . The product ab is required to satisfy the following conditions : ( i ) a ( bc ) = ( ab ) c , a , b , c e G ; ( ii ) there is an element e in G , called the identity or the unit of G , such ...
Page 40
... element a in a ring is said to be idempotent if x2 x and to be nilpotent if x = 0 for some positive integer n . A Boolean ring is one in which every element is idempotent . The identity a + 0 , or equi- -x , holds in every Boolean ring ...
... element a in a ring is said to be idempotent if x2 x and to be nilpotent if x = 0 for some positive integer n . A Boolean ring is one in which every element is idempotent . The identity a + 0 , or equi- -x , holds in every Boolean ring ...
Page 335
... element x which is in b but not a is an upper bound for a . It will first be shown that W satisfies the hypothesis ... element of coa and let y be any other element of c . If ye be Wo , we have either b a or a ≤ b . If b≤a , then ...
... element x which is in b but not a is an upper bound for a . It will first be shown that W satisfies the hypothesis ... element of coa and let y be any other element of c . If ye be Wo , we have either b a or a ≤ b . If b≤a , then ...
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