Exercise: The range of Compact Operators

Question

Exercise:

Suppose $K:X\to Y$ is compact operator.

1. Show that $K(X)\subseteq Y$ is separable
2. 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 $Y$.

Answer 1

Hint: A compact space is separable. Write $X=U_nB(0,n)$ $K(B(0,n)$ is separable since it is relatively compact, $\bigcup_nK(B(0,n))=K(X)$ is separable since it is the union of separable spaces.

For your second question let $(v_n\neq 0)$ be a dense family in $Y$, write $w_n={{v_n}\over{\|v_n\|}}$. Consider $K:l^1\rightarrow Y$ defined by $K(e_i)={w_i\over i}$