# Exercise: The range of Compact Operators

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 $Y$.

• What examples of Banach spaces do you know? – user99914 Jul 23 '16 at 17:38
• For separable Banach spaces I know: $L^p$, $Y$ s.t. $dim(Y)<\infty$ and $c_0$ (the space of sequence converging to zero). – MorganeMaPh Jul 23 '16 at 17:44
• For your second question, a recent question is very helpful. – user99914 Jul 23 '16 at 17:50
• thank you @ArcticChar I just checked the question you linked me but I don't get why it should be dense in $Y$? – MorganeMaPh Jul 23 '16 at 17:58
• Ah great! I got it, thank you! – MorganeMaPh Jul 23 '16 at 18:01

## 1 Answer

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 $K(e_i)={w_i\over i}$