Assume that $V$ is a Hilbert space. Let $\{e_n\} $ be an orthonormal basis. Let $T$ be such that $\{Te_n\} $ is also an orthonormal basis. Show that $T$ is unitary.
I have tried many ways to proof this fact, I know that the opposite is true, so that if $\{e_n\} $ is orthonormal basis and $T$ is unitary then $\{Te_n\} $ is an orthonormal basis too. I tried to prove it by using the fact that $\{e_n\}$ is an orthonormal basis if and only if $$ \forall_v v=\sum (v, e_n) e_n$$ and by knowing that $T$ is unitary if and only if $T$ is an isomorphism. But I tried many combinations of those facts and the definitions, but none of the approaches have worked.
Hint: To show that $T$ is unitary, it suffices to show that it is isometric and surjective. In the comments above, there is a hint that shows how you can show that $T$ is isometric. Next, recall that the image of an isometry is closed. Hence, to show surjectivity, all you need to do is show that the image is dense as well. Why is this clear?