Leray-Schauder fixed point theorem from Gilbarg and Trudinger book is quoted below. I do not understand remark below this theorem. Could you explain?
Theorem 11.2 from this text is Schauder fixed point theorem says: Every continuous mapping of closed convex subset $X$ of Banach space into itself such that $T(X)$ is precompact has fixed point. Compact operator between $A$ and $B$ is a operator which maps bounded sets in $A$ to precompact sets of $B$. I think I understand proof but I don't understand the remark, especially the second thesis.