I know that it would be true if $X$ and $Y$ both would be Banach spaces AND assuming that $T$ maps bounded and closed subsets of $X$ onto closed subsets of $Y$.
But what happens in the case that $X$ is just a normed vector space, $Y$ is Banach and one does NOT have the assumption about $T$?
My intuition says that in this case the range is not closed. But is there any example where I can see it clearly?
