Is it true that if $A$ is a convex set in a normed linear space $V$ , then the closure of $A$ is also convex ? (I know that the interior is convex )
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If $A$ is convex, then the convex hull of $A$ is its closure. – Surb Jan 13 '15 at 14:40
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@Surb: But if $A$ is convex , then the convex hull of $A$ is $A$ and if your claim is true,then $Conv(A)=A=\bar A $ , so every convex set becomes closed which is not true for example $A:=(0,1)$ – Jan 13 '15 at 14:45
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Yes. Let $x,y$ in the closure and $z = \alpha x + (1-\alpha)y$ with $0\leq \alpha\leq 1$. The point $x$ (resp. $y$) is a limit of a sequence $(u_n)_n$ (resp. $(v_n)_n$) of points of $A$, and $z$ is limit of the sequence $(w_n)_n$ with $w_n = \alpha u_n + (1-\alpha) w_n \in A$ by convexity of $A$. Therefore $z$ is limit of a sequence of points of $A$, and is in the closure of $A$.
Olórin
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