Suppose $K:X\to Y$ is compact operator.
- Show that $K(X)\subseteq Y$ is separable
- Assume $Y$ is a separable Banach space. Find a Banach space $Z$ and a compact operator $K:Z\to Y$ s.t. $K(Z)\subseteq Y$ is dense
Can someone help me to solve this exercise? Of course I know, that the image of the closed unit ball under $K$ is compact. Also I know that what to show is that it exists a countable dense subset $M$ in $Y$.