Analisis funcional, Teoria y aplicaciones by Hanm Brezis

By Hanm Brezis

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Ng (1 - PROOF: Fix f:) (VjnF j ) <;;: V j + l <:: Vj . •• ) arbitrarily but so that = r-T(l - f:j) converges and is positive. Let VI E U(F) 1 (1,1) holds. o n WE U(R) . Then (II, 1) cond"('- and (II,j) n any V3E U(F) f: = f: l ). Next we choose (with f: = f: 2 ), etc. Set V n j V. J be such that (I, 2) satisfying and and (I, 3) v. •• This . together with the second inclusion in n(VjnF j ) <;;: V R(\V = Rn U k (j = 1,2, ... e. y d 0 4 e. d F E(LF). o n4 : R VEU(F). From (I,j) we (V nF k ) <;;: W, and the lemma is proved.

Lau S. tion. R of R* = - x + R. U(F ) n This such that (n=1 , 2 , ... ) . V E U (F) and V n R* such that a nd F = 0. set VI nR~ = 0 . Fl being r e - 2 F2)) is O(F l , Fi)-compact, hence a lso O(F 2 , F )-compact. ve. 0 = n flexive, H J v n( R* n V (3) Rj = R*n F j . , F . We may assume that 01- R*, where will foll ow if we can exhibit a seque nce of V ' be the reflexive Banach spaces such that F " R. It suffices to show that Indeed, then for Rl Fn' Hence by (M) and thus also in n Consider a sequentially closed subspac e Sl <;;; S2 <;;; x c 1 ose d '~n v Le.

N i s in be. 6pa ce. 6pac e. d . 6pace. d. e. 6 . n F F. J F = lim. ind F .. Let J E F be. tive. the. lau S. tion. R of R* = - x + R. U(F ) n This such that (n=1 , 2 , ... ) . V E U (F) and V n R* such that a nd F = 0. set VI nR~ = 0 . Fl being r e - 2 F2)) is O(F l , Fi)-compact, hence a lso O(F 2 , F )-compact. ve. 0 = n flexive, H J v n( R* n V (3) Rj = R*n F j . , F . We may assume that 01- R*, where will foll ow if we can exhibit a seque nce of V ' be the reflexive Banach spaces such that F " R.

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