Linear Operators: General theory |
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Page 136
Nelson Dunford, Jacob T. Schwartz. 6 LEMMA . There is a uniquely determined smallest field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all ...
Nelson Dunford, Jacob T. Schwartz. 6 LEMMA . There is a uniquely determined smallest field and a uniquely determined smallest o - field containing a given family of sets . PROOF . There is at least one field , namely the field of all ...
Page 455
... determined by GF . - We can even assert that each continuous linear functional ƒ can be included in a denumerable self - determined set G of functionals . For , if f is determined by the denumerable set G1 , let each functional in G1 be ...
... determined by GF . - We can even assert that each continuous linear functional ƒ can be included in a denumerable self - determined set G of functionals . For , if f is determined by the denumerable set G1 , let each functional in G1 be ...
Page 657
... determined new state y . Since y is uniquely determined by a and t , a function & on S to S is defined by the equa- tion y = ( x ) . The flow is assumed to have the property that ( i ) $ t ( $ s ( x ) ) = $ t + s ( X ) , for all points ...
... determined new state y . Since y is uniquely determined by a and t , a function & on S to S is defined by the equa- tion y = ( x ) . The flow is assumed to have the property that ( i ) $ t ( $ s ( x ) ) = $ t + s ( X ) , for all points ...
Contents
A Settheoretic Preliminaries | 1 |
10 | 30 |
Three Basic Principles of Linear Analysis | 49 |
Copyright | |
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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