Let $A: X\rightarrow Y$ be a linear operator between two Banach spaces $(X, ||\cdot||_X)$ and $(Y, ||\cdot||_Y)$. Show that $A$ is continuous if and only if $\ker(A)\subset X$ is closed.
Possible duplicates:
Showing that $\ker T$ is closed if and only if $T$ is continuous. or
$T$ is continuous if and only if $\ker T$ is closed
I'm trying to prove the statement without using the quotient space and without supposing $Y= \mathbb{R}$, as done in the two url before.
First direction is clear, but I have some trouble to prove the converse.
Any suggestions? Thanks in advance!
Best Answer
You cannot prove it, since it is not true. If it were true, then any injective operator between Banach spaces would be continuous.
Take any two infinite-dimensional Banach spaces with the same dimension (as vector spaces). Then there is a linear isomorphism $L$ between them, whose kernel is closed, since it is $\{0\}$. But, since they are not isomorphic, either $L$ is not continuous or $L^{-1}$ is not continuous. And $\ker L^{-1}=\{0\}$ too.