If $Y$ is a subset of topological vector space $X$ and is compact and convex show that $\overline{Y^\circ} = \overline{Y}$ and $\overline{Y}^\circ = Y^\circ$.
I tried this way but I am not sure:
$Y$ is compact so $Y = \overline{Y}$. Then it follows that $Y^\circ = (Y)^\circ = (\overline{Y})^\circ$. And for another I stuck so could some help.
One more thing: What happens if $Y$ is not compact and convex the proof is true for this case too.
This is not a homework question.
Of course, as previously mentioned in the above comments, the condition that $Y^\circ$ is non-empty is needed. In this case $Y$ need not be compact. You can find the proof in Holmes's "Geometric functional analysis ..." book on p. 59.