I would like to ask a counterexample for the classical theorem in functional analysis: the open mapping theorem in the case that $Y$ is Banach, but $X$ is not Banach to show that the completeness of X is crucial.

In details, find a continuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $Y$ is Banach but $T$ is not open.

If we can construct this, we could get an interesting example: there exists a bijective linear (contiuous) mapping between two normed space $X$ and $Y$, and only one of them is Banach. The counterexamples for the case when $Y$ is not Banach is simple, but I didn't come up if I need $X$ is not Banach and $Y$ is Banach. Thanks!

research level. It is almost perfectly suited for Math Stack Exchange (I think), since the basic tools to find the required example (like a Hamel basis, the existence of unbonded linear functionals etc.) are sufficient. In some sense it is an exercise everybody should solve, but not get solved within one hour on MO. I am sure that a crowd of interested students would have been able to answer this question quickly. $\endgroup$ – Andreas Thom Nov 4 '10 at 7:16