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 9. 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 9. 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 1 = K ( s , t ) f ( t ) dt , ds 0 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 1 = K ( s , t ) f ( t ) dt , ds 0 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 B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ