TITLE: Definition of Compact Mapping
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: The first is true and the second is the definition of a Compact operator (different from a mapping at all and necessarily linear) on a Banach space by this changes:
Let $X$ be a Banach space. Then a linear operator ( necessarily linear) $f:X\to X $ is compact if the closure of $f(Y)$ is relatively compact whenever $Y\subset X$ is bounded.