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 X1 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 = O is ...
... linear map x → x of X2 onto X1 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 = O is ...
Page 490
... linear map from L , [ 0 , 1 ] , p > 1 , to L , [ 0 , 1 ] has the form g ( s ) d ds Jo 1 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 Jo 1 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 continuous linear converges convex set Corollary countably additive DEFINITION disjoint Doklady Akad E₁ element exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear functional 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 Russian 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