Re: Functional analysis: space of bounded varation separable?

From: Robert Israel (israel_at_math.ubc.ca)
Date: 11/09/04

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    Date: 9 Nov 2004 02:04:48 GMT

    In article <418FD35B.5090105@web.de>,
    Jannick Asmus <jannick.news@web.de> wrote:
    >On 08.11.2004 18:35, Robert Israel wrote:
    >> In article <cmnoqv$a4l$1@online.de>,
    >> Jannick Asmus <jannick.news@web.de> wrote:

    >>>Is it true that the space of real valued functions on R of bounded
    >>>variation is separable?

    >> No, it isn't. For example, the indicator functions of intervals
    >> [a,b] form an uncountable family, and the total variation of the
    >> difference of any two of them is at least 2.

    >I need to think about your argument, because, meanwhile, I found out:
    >BV[0,1] is the dual of C[0,1] (via the pairing induced by the
    >Riemann-Stieltjes-integral). Since C[0,1] is a separable Banach space
    >(Stone-Weierstrass), so is its dual BV[0,1].

    No, the dual of a separable Banach space is not necessarily separable.
    C[0,1] is one good example of this, l^1 (the space of summable sequences,
    whose dual is identified with l^infinity) is another.

    Robert Israel israel@math.ubc.ca
    Department of Mathematics http://www.math.ubc.ca/~israel
    University of British Columbia Vancouver, BC, Canada

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