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 , xe R ; ( b ) ( 0 ) I ÷ R. 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 , xe R ; ( b ) ( 0 ) I ÷ R. 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 197
... on R to X such that lim fn ( r ) —f ( r ) | v ( o , dr ) = 0 . ∞4u R It is seen , by using Theorem 9 , Lemma 11 , and Lemma 2.18 , that the function G ( s ) = f ( s , · ) is in Ĩ , and that | Fn — G | = √s | Fn ( 8 ) —G ( s ) | v ( μ ...
... on R to X such that lim fn ( r ) —f ( r ) | v ( o , dr ) = 0 . ∞4u R It is seen , by using Theorem 9 , Lemma 11 , and Lemma 2.18 , that the function G ( s ) = f ( s , · ) is in Ĩ , and that | Fn — G | = √s | Fn ( 8 ) —G ( s ) | v ( μ ...
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