Linear Operators: General theory |
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Page 38
... of R is called a right ideal of R if it has the additional properties ( a ) Ix CI , x ‹ R ; ( b ) ( 0 ) 1R . The definition of left ideal is similar . If I is both a right and a left ideal of R it is called a two - sided ideal . The ...
... of R is called a right ideal of R if it has the additional properties ( a ) Ix CI , x ‹ R ; ( b ) ( 0 ) 1R . The definition of left ideal is similar . If I is both a right and a left ideal of R it is called a two - sided ideal . The ...
Page 39
... of R has an inverse , so R is a field . A right ( left , or two - sided ) ideal in a ring R is called a maximal right ( left , or two - sided ) ideal , if it is contained in no other ideal of the same type . If R contains a unit element ...
... of R has an inverse , so R is a field . A right ( left , or two - sided ) ideal in a ring R is called a maximal right ( left , or two - sided ) ideal , if it is contained in no other ideal of the same type . If R contains a unit element ...
Page 586
... and R ( λ ; T ( μ ) ) is an analytic function of μ for each λ € U. μ PROOF . By Lemma 3 , there is a d1 such that if ... to R ( λ ; T ) uniformly for λ in B if | T1 — T | is small . Thus , for some positive d≤ d1 , f ( T ) −f ( T ) 1 ...
... and R ( λ ; T ( μ ) ) is an analytic function of μ for each λ € U. μ PROOF . By Lemma 3 , there is a d1 such that if ... to R ( λ ; T ) uniformly for λ in B if | T1 — T | is small . Thus , for some positive d≤ d1 , f ( T ) −f ( T ) 1 ...
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