It was proved by Schauder himself

J. Schauder, Über die Umkehrung linearer, stetiger Funktionaloperatoren, Studia Math. 2 (1930), 1–6.

see p. 6. It also 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.

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.