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 æ in a ring is said to be idempotent if x2 = x and to be nilpotent if xn 0 for some positive integer n . A Boolean ring is one in which every element is idempotent . The identity x + x = 0 , or equi- valently x = -x , holds in ...
... element æ in a ring is said to be idempotent if x2 = x and to be nilpotent if xn 0 for some positive integer n . A Boolean ring is one in which every element is idempotent . The identity x + x = 0 , or equi- valently x = -x , holds in ...
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 ca 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 ca 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 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite 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 TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ