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 x 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 x + x = 0 , or equi- valently x = -x , holds in ...
... element x 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 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
Preliminary Concepts | 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 ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f 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 S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ