I was reading around the other day and came across the term "compact mapping". After googling, I saw the following two definitions:
Let $X$ be a topological space. Then a mapping $f:X \to X$ is compact if $f^{-1}(\{x\})$ is compact for every $x \in X$.
Let $X$ be a Banach space. Then a mapping (not necessarily linear) $f:X \to X$ is compact if the closure of $f(Y)$ is compact whenever $Y \subset X$ is bounded.
Are these definitions equivalent if $X$ is a Banach space? If not, what is the usual meaning in the context of Banach spaces? For example, Schaefer's Fixed Point Theorem states
If $X$ is a Banach space and $f:X \to X$ is a continuous and compact mapping such that $\{x \in X: x = \lambda f(x) \mbox{ for some } 0 \leq \lambda \leq 1\}$ is bounded then $f$ has a fixed point.
Which definition is meant? Sorry if I am missing something obvious here.