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Page 244
... Hence the sequence { y } converges to the vector [ 1 , in En . 1 1 , . . . , n , • . • xn ] LEMMA . A finite dimensional normed linear space is complete and hence it is a B - space . i = 1 - - 1 and yn = T - 1 - n PROOF . Let { b1 ...
... Hence the sequence { y } converges to the vector [ 1 , in En . 1 1 , . . . , n , • . • xn ] LEMMA . A finite dimensional normed linear space is complete and hence it is a B - space . i = 1 - - 1 and yn = T - 1 - n PROOF . Let { b1 ...
Page 441
... hence a compact , subset of co ( Q ) . Hence Ki - co ( Q ) = co ( K1U ...... . U K2 ) = co ( K1U ... UKn ) , Ο , by an easy induction on Lemma 2.5 . It follows readily that Р has the form p Σak , a , ≥0 , Σa , = 1 , k , e K ; and ...
... hence a compact , subset of co ( Q ) . Hence Ki - co ( Q ) = co ( K1U ...... . U K2 ) = co ( K1U ... UKn ) , Ο , by an easy induction on Lemma 2.5 . It follows readily that Р has the form p Σak , a , ≥0 , Σa , = 1 , k , e K ; and ...
Page 485
... Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the * , *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in X , X ...
... Hence T * is weakly compact . Conversely , if T * is weakly compact , it follows from Lemma 7 that T ** is continuous relative to the * , *** topologies in X ** , Y ** , respectively . If S , S ** are the closed unit spheres in X , X ...
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