# Proof of the Schauder Lemma

Schauder's Lemma in functional analysis states the following:

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.

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 $\overline{A(E)}=F$.

• If the space $E$ is Banach, then there is a simple proof, which exploits the notion of an $\infty$-convex set, see matstud.org.ua/texts/1999/11_1/11_1_083-084.pdf – Taras Banakh Jul 17 '17 at 19:31
• I don't see a simple way to adapt that argument though. – D. Wynter Jul 17 '17 at 20:34
• If $E$ and $F$ are Banach, then its unit ball $U$ is $\infty$-convex, and so is its image $A(U)$. By the assumption, the closure of $A(U)$ in $F$ contain some open ball $V$ of $F$. Then $V\cap A(U)$ is dense $\infty$-convex set in $V$ and hence coincides with $V$ by the Static Property proved in that paper. Therefore, $A(U)$ contains the ball $V$ and $A(U-U)$ contains the ball $V-V$ centered at zero, which means that the operator $A$ is open. This gives the proof of the "Banach" case. I do not know if this argument can be adapted to the "Frechet" case. – Taras Banakh Jul 18 '17 at 1:50

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):

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.

As far as I remember this can also be found in Bourbaki.

It follows from

I. M. Dektjarev, A closed graph theorem for ultracomplete spaces, Dokl. Akad. Nauk. Sov., 154 (1964), 771–773 (in Russian).

Below you can find some references. They do not give the result directly in the form you want but instead give more general results where $E$ is assumed to be a Pták space (= B-complete = fully complete) which then gives the result you want by the implication Fréchet $\Rightarrow$ Pták.