Proof of the Schauder Lemma

It follows from

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

See also discussion on p. 5 here.

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.

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.

[1] J. Horváth: Topological Vector Spaces and Distributions, Theorem 3.17.2, p. 296.

[2] H. Jarchow: Locally Convex Spaces, Theorem 9.7.1, p. 186.

[3] H.H. Schaefer: Topological Vector Spaces, Theorem 8.3, p. 163.