Schauder's Lemma in functional analysis states the following:



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> Let $E$ and $F$ be metrizable locally convex topological vector spaces, and let $E$ be Fréchet. Then if the linear continuous map $A:E\to F$ is *nearly open*, that is to say, for any neighborhood of the origin $U\subset E$ we have that $\emptyset\ne\operatorname{Int}(\overline{A(U)})\subset F$, then $A$ is surjective and open.
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However, I've been unable to find a reference for a proof of this, and I haven't managed to figure out why it is true either. Either a proof or a reference to a proof would be much appreciated. The most I've been able to show is that $\overline{A(E)}=F$.



Proof of the Schauder Lemma
The result has not much to do with the linear structure. In the book *Introduction to Functional Analysis* of Meise and Vogt you find a version for metric spaces (Lemma 3.9):



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> Let $X$ and $Y$ be metric spaces; $X$ be complete. Let $f:X\to Y$ be continuous and assume that 
>  for every $\varepsilon>0$ there exists a $\delta>0$, such that for all $x\in X$, $\overline{f(U\_\varepsilon(x))} \supset U\_\delta(f(x))$.
>  Then the map $f$ is open.
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As far as I remember this can also be found in Bourbaki.


