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Let $V$ be a real locally convex space. Let $F : V \to R$. Are the following equivalent?

(a) $\{ u \in V : F(u) \leq a \}$ is closed for any $a \in R$.

(b) $\liminf_{n} F(u_n) \geq F(u)$ whenever $u_n \to u$.

(I see (a) $\Rightarrow$ (b) but not the converse.)

user66081
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    When $V$ is metrizable, yes. Suppose (a) fails. Then there is $a$ such that ${ u \in V : F(u) \leq a }$ is not closed. This means there is a sequence $(u_n)$ of elements of this set with limit $u$ outside of this set. Hence, $\liminf_n F(u_n)\le a <F(u)$.... I suspect the equivalence may fail for non-metrizable spaces since sequences do not tell the whole story. –  Dec 06 '14 at 03:48

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