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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!

userr777
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1 Answers1

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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.

José Carlos Santos
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