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 = a 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 + x = 0 , or equi- valently x -x , holds in ...
... element a in a ring is said to be idempotent if x2 = a 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 + 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 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