From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Compact of L_infinity
Date: 5 Oct 1999 13:27:05 -0400
Newsgroups: sci.math
In article <37FA0FCB.7E1EA624@inpg.fr>,
Nicolas Marchand wrote:
:Hi,
:
:What are the compact subsets of L_infinity ?
Theoretically, L_infinity is isomorphic (as a Banach lattice) to C(Y) for
a compact Hausdorff Y. So, a form of Arzela-Ascoli theorem can be put
together, but the formulation would be "excessively cumbersome", as
Dunford-Schwartz note in a similar situation (IV.15). This Y will be
extremely disconnected, and non-metrizable in the interesting situations.
>Can i extract a convergent series from a series (f_n) of essentially
>bounded functions (same bound) ?
Not always (and you probably mean "sequences"): in L_infinity(0,1), take
any list of characteristic functions of open intervals with rational
endpoints; the distance between any two members is exactly 1.
Cheers, ZVK(Slavek).
==============================================================================
From: "G. A. Edgar"
Subject: Re: Compact of L_infinity
Date: Tue, 05 Oct 1999 14:07:22 -0400
Newsgroups: sci.math
In article <7tdcd9$i3b@mcmail.cis.McMaster.CA>, Zdislav V. Kovarik
wrote:
> This Y will be extremely disconnected
also extremally disconnected
--
Gerald A. Edgar edgar@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)