I have a - maybe slightly stupid - question about the Schauder-Fixed-Point Theorem.
The formulation I have in mind is:
Let $A$ be a closed, convex, nonempty subset in a Banach space $(X,\|\cdot\|)$, and $T \colon A \rightarrow A$ be a continuous, compact mapping. Then $T$ has a fixed point.
So my question is: When the assumptions are speaking about a subset, does it mean $A \subseteq X$ or $A \subsetneq X$?
Specifically, could I take $X = L^2(\Omega)$ and $A = L^2(\Omega)$, since then $A \subseteq X$ is closed in $X$, convex and nonempty, and conclude for a continuous map $T \colon L^2(\Omega) \rightarrow H^1(\Omega)$ by compact embedding $H^1(\Omega) \hookrightarrow L^2(\Omega)$ the existence of a fixed point in this situation?
Thanks a lot!