Find a space whose dual does not separate points

Question

I read about the fact that for a locally convex topological vector space $X$, its dual $X^*$ separates points, i.e. for any $x\neq y$ in $X$, $\exists f \in X^*$ such that $f(x)\neq f(y)$.

Could you help me to find a non locally convex topological vector space such that its dual does not separate points? Thanks

Do you know any non-locally-convex topological vector spaces? If so, which ones? — Daniel Fischer, Oct 02 '14 at 20:17

score 11 · Accepted Answer · answered Oct 02 '14 at 20:22

11

You can equip $L^p([0,1])$ with the metric $\|f-g\|^p_p = \displaystyle \int_0^1 |f(t) - g(t)|^p \, dt$ if $0 < p < 1$. It is a (fairly) well-known result due to Mahlon M. Day that the only bounded linear functional on this space is the zero functional. In particular, the space is not locally convex.

answered Oct 02 '14 at 20:22

Umberto P.

52,165

Thanks! I've seen this example several days ago. Then I forget it – Petite Etincelle Oct 02 '14 at 20:28
Interesting. I know next to nothing about non-Banach functional analysis (not to mention non-locally convex), so it's kind of surprising to me. +1. :) – tomasz Oct 02 '14 at 21:51

Find a space whose dual does not separate points

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