Finding extreme points of closure of convex hull

Question

Let H be a infinite dimensional Hilbert space with orthonormal basis $(e_n)_{n\geq 1}$. Let $f_N=N^{-\frac{1}{2}}\sum_{n=1}^Ne_n$ for all $N\geq 1$ and let K be the norm closure of the convex hull of $\{f_N : N\geq 1\}$. I need to show that the extreme points of K are $\{0\} \cup \{f_N:N\geq 1\}$.

I have already shown that the $f_N$'s are extreme points, but I do not know how to show that 0 is an extreme point.

I also can't figure out how to show that these are the only extreme points. I have shown that K is weakly compact, so following the converse to Krein-Milman i know that $\text{Ext}(K)\subset \overline{\{f_N:N\geq 1\}}^\tau$, where $\tau$ is the weak topology. But I do not know how to show that $\overline{\{f_N:N\geq 1\}}^\tau\subset \{0\} \cup \{f_N:N\geq 1\}$. One way could be to consider all possible nets in $\{f_N:N\geq 1\}$, but this has not yet proven fruitful. Any help is much appreciated

Because $f_N$ converges to 0 in the weak topology. edit:Someone asked why $0\in K$ — Caspar, Jan 12 '20 at 02:02
That implies $0$ is in the weak closure of the CH of the $f_N$, but not that its in the norm-closure. What am I missing? — kimchi lover, Jan 12 '20 at 02:04
The norm closure and the weak closure are the same for convex sets — Caspar, Jan 12 '20 at 02:05
Thanks! (Doh, head-slap.) OK: Every convex combination $g$ of the $f_N$ must have $\langle g, e_1\rangle>0$. So $0$ can't be a non-trivial combo of non-zero elements of $K$. — kimchi lover, Jan 12 '20 at 02:15
Hmm, but the elements in $K$ are the limits of convex combinations. So I'm not sure how to adapt your argument. — Caspar, Jan 12 '20 at 02:42

kimchi lover · Accepted Answer · 2020-01-14T13:38:29.873

Since $f_n$ converges weakly to $0$, we have $0$ is in the weak closure of the convex hull of $\{f_n\}$, and hence by a standard result, $0$ is contained in the strong closure of the convex hull, that is, $0\in K$.

Let $E$ denote the extreme points of $K$.

If $0$ were not in $E$ one could write $0$ as a non-trivial convex combination of distinct elements of $K$, that is, $0=pu+qv$ for $u,v\in K$ for some $u\ne v$, and for some reals $p,q>0$ for which $p+q=1$. But each element of $K$ is contained in the intersection of the closed convex half-spaces $H_k=\{u\in H : \langle e_k,u\rangle\ge 0\}$. So for each $k$ we have $u,v\in H_k$. But that means $0=p\langle e_k,u\rangle +q\langle e_k,v\rangle$. Since $0$ is an extreme point of the set of non-negative reals, this means that both $\langle e_k,u\rangle$ and $\langle e_k,v\rangle$ vanish. Since the $e_k$ form a basis, this implies $u=v=0$, contrary to supposition.

Let $F=\{f_n\}\cup\{0\}$; we have just seen that $F\subseteq E$. Since we know $0\in K$, however, we see that the closed convex hull of $F$ is equal to $K$. If there was an extreme point $e\in E$ not in $F$, then there would be a contradiction: one could strictly separate $e$ from the closed convex hull of $F$ with a hyperplane. But $K$ is at once the closed convex hull of $F$ and (by the Krein-Milman theorem) of $E$, so this cannot be.

Or, more directly, using Proposition 1.5 in Phelps's Lectures on Choquet Theory, p.6

Suppose that $X$ is a compact convex subset of a locally convex space, that $Z\subset X$, and that $X$ is the closed convex hull of $Z$. Then the extreme points of $X$ are contained in the closure of $Z$.

where the LCS is $H$ with the weak topology, where $\{f_n\}$ plays the role of $Z$, $F$ plays the role of the closure of $Z$, and $K$ plays the role of $X$.

And how would you show that $a=\sum_{k\geq 1}\pi(k)f_k$? Dominated convergence don't seem to work — Caspar, Jan 12 '20 at 19:13
I think I have another idea that solves it. ${0}\cup {f_N : N \geq 1}$ is weakly closed because it is weakly compact. — Caspar, Jan 12 '20 at 21:33
@Caspar how do you know ${0 } \cup { f_N : N \geq 1 }$ is weakly closed/compact? I think it's right, but I don't see how to show it — averagemonkey, Jan 14 '20 at 14:20
@averagemonkey You can show it directly using the definition of compactness. Take a open cover and find a finite sub-cover. Use that $f_N$ converges weakly to 0. — Caspar, Jan 14 '20 at 15:45

Finding extreme points of closure of convex hull

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