# Read e-book online Advances in Mathematical Economics PDF

By Gérard Debreu (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

ISBN-10: 4431658955

ISBN-13: 9784431658955

ISBN-10: 4431658971

ISBN-13: 9784431658979

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**Additional resources for Advances in Mathematical Economics**

**Sample text**

E H has the WTP). (3) Given any sequence (fn) in H, there exists a subsequence (gn) of(fn) such that (l{llh nll

3. 6 provide the characterizations of weak compactncss in L3c (p,) in terms of weak Komlos convergence. See [24, 31, 32] for details. 4. The characterizations of weak compactness and weak conditionally compactness in the space Lk, [E] (E being a Banach space) of scalarly integrable E'-valued functions via WTP appear in a forthcoming paper by Benabdellah-Castaing [11]. It is worthy to pose the problem of characterizing the Banach X by the previous modes of convergences. The following result gives a positive answer to this question.

Castaing, M. Guessous for each subsequence (wt) = (v rn [). Since (u n ) converges (J(L1,LOO) to u, so is (v m [). It follows that \-I vp E l"n (* ) ;;;,L"j=l ep ' VrnJ ThT 1'1, 17(L',L=) ---+ (* ) ep ' U . Whence we get \;/p E N, 'Pp = (e;', U), thus completing the proof. D Remarks. 4. (1) If X is reflexive, L~(J-l) has the WTP (see, for example, [9, 10]). (2) If (u n ) is a bounded sequence in L~(J-l) and Un{un(w)} is ballrelatively weakly compact for every wEn, that is its intersection with any closed ball is relatively weakly compact, then (u n ) has the WTP.

### Advances in Mathematical Economics by Gérard Debreu (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

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