/sci/ - Science & Math

Consider a (weakly) closed set [math]X[/math] in a Hilbert space, and define [math]\overline{\mathrm{co}}(X)[/math] to be the weak closure of its convex hull, with
[math]\mathrm{co}(X) = \{ \sum_{i=1}^n a_i x_i : a_i \geq 0,\, \sum_{i=1}^n a_i = 1,\, x_i \in X \}[/math].

Can any element [math]y \in \overline{\mathrm{co}}(X)[/math] be written as [math]y = \sum_{i=1}^{\infty} a_i x_i[/math] for some choice of [math]x_i \in X[/math] and nonnegative [math]a_i[/math] with [math]\sum_{i=1}^{\infty} a_i = 1[/math]?

I remember reading somewhere that this is not possible in a general Banach space, i.e. there will not always exist a countable decomposition of any [math]y \in \overline{\mathrm{co}}(X)[/math] into elements of [math]X[/math], but one needs to consider more general (continuous) measures. Not sure if this can be done in a Hilbert space?