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 -> Y which 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!

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!