Image of a convex set under a compact map

Question

How do I prove that $T(\overline\Omega)\subset\overline\Omega$, where $$T:\overline\Omega\rightarrow X$$ is compact (X Banach, $\Omega\subset X$ convex and bounded) and $$T(\partial\Omega)\subset\Omega$$

Is this even true for any convex subset of X?

Answer 1

First $\overline{\Omega}$ is a convex set, and you can show pretty easily that $T\big( \overline{\Omega} \big)$ is convex. Furthermore $\overline{\Omega}$ is bounded and therefore by compactness of $T$ we know therefore that $T \big( \overline{\Omega} \big)$ is compact (can be seen by sequential compactness). By the Krein-Milman theorem we know that:

$ \overline{conv} \Big( Ext \big( T\big( \overline{\Omega} \big) \big) \Big)= T\big( \overline{\Omega} \big) $

Then it just remains to show that $Ext \big( T\big( \overline{\Omega} \big) \big)\subseteq \overline{\Omega}$. But since for a given set $A$, we know that $Ext(A)\cap int(A)=\emptyset$, and $\partial A = \overline{A}\setminus int(A)$ we know then that $Ext(A)\subseteq \partial A$. Applied to the problem hand it specifically gives us that:

$ Ext\Big( T\big( \overline{\Omega} \big) \Big) \subseteq T\big( \partial \Omega \big) $

And finally by the assumption $T\Big( \partial \Omega \Big) \subseteq \Omega$, and therefore $Ext \big( T\big( \overline{\Omega} \big) \big)\subseteq \Omega$ as needed.