definition of shauder fixed point theorem

Question

I read the definition of Schauder fixed point theorem as follows. If $T:B\rightarrow B$ is an operator, defined on normed linear space $B$ into $B$ such that $T$ is continuous on $B,$ compact operator and for a nonempty closed bounded convex set $S\subset B,$ $T(S)\subset S$ then $T$ has a fixed point in $B.$ My questions are, even $T$ can be nonlinear operator? What is the meaning of compact operator here? $T(S)\subset S$ should be true for at least one $S?$ or for all $S?$ Further compact linear operator means, given a bounded set $S,$ $\overline{T(S)}$ is a compact set in $B.$ But whether this condition should be true for every bounded set $S?$ or for at least one $S?$

Answer 1

The hypotheses, as stated in the Wikipedia page, are rather different from what you wrote:

1. $T$ is a continuous mapping of a nonempty closed convex subset $S$ of a Hausdorff topological vector space into itself. (In particular it could be a normed linear space, but does not have to be)
2. $T(S)$ is contained in some compact subset of $S$.

The conclusion is that $T$ has a fixed point in $S$.

As for your questions:

$T$ is not assumed to be linear, or indeed to be defined anywhere other than $S$.

"Compact" appears in one place: $T(S)$ is contained in a compact subset of $S$.

There is only one $S$.