Linear Operators: General theory |
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Page 57
... linear one - to - one map of one F - space onto all of another has a continuous linear inverse . PROOF . Let X , Y be F - spaces and T a continuous linear one - to- one map with TX Y. Since ( T - 1 ) −1 T maps open sets onto open sets ...
... linear one - to - one map of one F - space onto all of another has a continuous linear inverse . PROOF . Let X , Y be F - spaces and T a continuous linear one - to- one map with TX Y. Since ( T - 1 ) −1 T maps open sets onto open sets ...
Page 58
... linear map x → x of X2 onto X is continuous . By Theorem 2 , it is a homeomorphism , and so T1 = T2 . Q.E.D. 6 DEFINITION . A family F of functions which map one vector space X into another vector space Y is called total if x = 0 is ...
... linear map x → x of X2 onto X is continuous . By Theorem 2 , it is a homeomorphism , and so T1 = T2 . Q.E.D. 6 DEFINITION . A family F of functions which map one vector space X into another vector space Y is called total if x = 0 is ...
Page 490
... linear map from L , [ 0 , 1 ] , p > 1 , to L , [ 0 , 1 ] has the form g ( s ) = d 1 ( * K ( s , t ) f ( t ) dt , ds Jo no satisfactory expression for the norm of T is known . No conditions on K ( s , t ) are known which are equivalent ...
... linear map from L , [ 0 , 1 ] , p > 1 , to L , [ 0 , 1 ] has the form g ( s ) = d 1 ( * K ( s , t ) f ( t ) dt , ds Jo no satisfactory expression for the norm of T is known . No conditions on K ( s , t ) are known which are equivalent ...
Contents
8 | 28 |
Algebraic Preliminaries | 34 |
Three Basic Principles of Linear Analysis | 49 |
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A₁ additive set function algebra Amer analytic arbitrary B-space ba(S Banach Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense differential disjoint Doklady Akad E₁ element equation exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism L₁ L₁(S Lebesgue Lemma Let f linear map linear operator linear topological space Math measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative normed linear space o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory TM(S topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ