Recently I learned in class Caratheodory's convex hull theorem which suggests that in a finite dimentional (n) space every x can be written as a convex-combination of maximum (n+1) vectors.
I was trying to think why this would not hold for infitine dimention. So I was looking for counter examples, i.e an infinite dimentional vector space and a subset A such as for every natural number n there is x $\in$ conv(A) that cannot be written as a convex-combination of n vectors in A.
The main problem that I encountered was that the only infinite dimentional spaces I know are ($L^p$, $\ell ^p$, and Sobolev space [which is ugly]). The seocnd was that I don't know how to "imagine" the convex hull of any (not convex) set in those spaces.
I'd love if you could guide me in order to learn how to find the convex hulls of such spaces, and maybe give me some more examples of infinite dimentional spaces.
Thanks!
Consider $A=\{0_{\ell^2}, e_1, e_2, \ldots\}\subset \ell^2$, where $e_i$ are the usual sequences with $1$ in the $i$-th position and $0$'s elsewhere. Take $x_n=(2^{-1}, \ldots, 2^{-n}, 0, 0, \ldots)$, then $$ x_n= (1-\sum_1^n 2^{-i})0_{\ell^2} + \sum_1^n 2^{-i}e_i, $$ so that $x_n\in \text{conv}(A)$, but clearly can't be represented by less than $n$ elements in $A$.