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 454
... determined by G , then g ( p ) = g ( q ) , ge G , implies that f ( Tp ) = f ( Tq ) , ƒ € F. Each continuous linear functional ƒ is determined by some de- numerable set of functionals G { m } . For , by IV.6.9 , the scalar function f ...
... determined by G , then g ( p ) = g ( q ) , ge G , implies that f ( Tp ) = f ( Tq ) , ƒ € F. Each continuous linear functional ƒ is determined by some de- numerable set of functionals G { m } . For , by IV.6.9 , the scalar function f ...
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 ...
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 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 exists f₁ finite dimensional function defined function f g₁ 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-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ