I have just been introduced to the concept of invertibility for bounded linear operators. Specifically, we defined a bounded operator $A$ to be invertible if there exists a bounded $A^{-1}$ which is its right and left inverse, i.e. $AA^{-1}=\mathrm{id}_{\mathrm{Im}A},A^{-1}A=\mathrm{id}_{\mathrm{Dom}A}$. So I was wondering: is the requirement of boundedness (or equivalently of continuity) of the inverse important? Or, is it asking more than is granted by the invertibility? The open mapping theorem states a continuous linear surjective operator between Banach spaces is an open map, and thus if it has an inverse, that inverse is necessarily continuous. So for Banach spaces, we could avoid requiring this continuity explicitly, as it is automatic. But what about non-Banach spaces?
So the question is: do there exist normed spaces $X,Y$ and operators between them which are linear and bounded but have un-bounded inverses?
Best Answer
Yes, unbounded inverses exist. Here is an example where $X$ and $Y$ are normed spaces.
For more examples, see Mohamed's answer here, where $X$ is actually Banach.
And for examples with $Y$ Banach, see this overflow post.