# Relation between Ranges of compact operators

I am reviewing functional analysis and getting stuck in this problem. Let $X,Y$ be two Banach spaces and $A,B\in L(X,Y)$. Prove that if $A$ is a compact operator and $R(B)\subset R(A)$ then $B$ is also a compact operator.

Can anyone give me some hints for this question ? Thank you very much.

• I think you need to assume that $B$ is bounded – Ben Grossmann Jun 26 '16 at 4:11
• Yes. B is a bounded linear operator – Omega Jun 26 '16 at 4:13

## 1 Answer

Endow $R(A)$ with the quotient norm having the unit ball $A(K_X)$ where $K_X$ is the unit ball of $X$. This is a Banach space and $\tilde B:X\to R(A)$, $x\mapsto B(x)$ is well-defined by assumption, linear, and has closed graph (because the norm of $Y$ gives a coarser norm on $R(A)$ which makes $B$ continuous). The closed graph theorem implies the continuity of $\tilde B$ so that $\tilde B(K_X) =B(K_X) \subseteq c A(K_X)$ for some $c>0$. Since $A(K_X)$ is relatively compact in $Y$ the same holds for $B(K_X)$.

• What is the quotient norm on R(A) #Jochen – Omega Jun 27 '16 at 23:58
• The quotient norm of $z\in R(A)$ is $\inf\lbrace \|x\|_X: A(x)=z\rbrace$. The open unit ball of the quotient norm is the image of the open unit ball of $X$. – Jochen Jun 28 '16 at 10:23