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Possible Duplicate:
The Space $C(\Omega,\mathbb{R})$ has a Predual?

Is their Banach space such that its dual is C[a b] - continuos functions on [a b] ?

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Related question is:

The Space $C(\Omega,\mathbb{R})$ has a Predual?

It seems that comment by Jonas Meyer to this question contains a negative answer to my question. However some extension of comment is welcome - "by the Krein-Milman theorem this implies that is not a dual space." - not very clear to me.

PS Surprising how can similar questions come to mind of different people during the same day ?:)

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Background:

in finite-dimensions: double dual space is isomorphic to itself. In infinite-dimensions it is not always true. In particular for space of continuous functions it is not true (as far as I remember). (See Double dual of the space $C[0,1]$ ).

Remark: Dual space of continuous functions is quite well understood - it is a space of measures with bounded variation. See e.g. http://regularize.wordpress.com/2011/11/11/dual-spaces-of-continuous-functions/

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    Regarding my comment, that was specific to the real case. Krein-Milman and Alaoglu imply that the weak*-closed convex set generated by the extreme points of the unit ball of a dual Banach space is equal to the unit ball, but for $C([a,b],\mathbb R)$, this would only give you constant functions in the unit ball. But as Matthew Daws says, Philip Brooker's answer covers this and more.2012-02-10

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