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 ...
... 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 ...
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 ds K ( s , t ) f ( t ) dt , no satisfactory expression for the norm of T is known . No conditions on K ( s , t ) are known which are equivalent to the ...
... linear map from L , [ 0 , 1 ] , p > 1 , to L , [ 0 , 1 ] has the form g ( s ) = - d ds K ( s , t ) f ( t ) dt , no satisfactory expression for the norm of T is known . No conditions on K ( s , t ) are known which are equivalent to the ...
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