Let $T:V\to V$ and two bases of $V$: $E = \{v_1, \ldots, v_n \}$ and $F = \{T(v_1), \ldots, T(v_n)\}$.
Prove: $E,F$ are orthonormal implies $T$ is unitary.
So basically we want to prove that for every $v\in V$:$$TT^*(v) = 1_v \iff TT^*(v)-1_v(v) = 0 \iff (TT^*-1_v)(v)$$
First of all, it's a well-known fact that $TT^*-1_v$ is conjugate to itself. If we proved that for every $v\in V$: $$\langle TT^*(v)-1_v(v), v\rangle = 0 $$
It would imply that $TT^*-1_v \equiv 0$ and we're done.
So I'm stuck at this point (not sure if even this is the right/optimial way)
We can restate this relationship $$\langle TT^*(v)-1_v(v), v\rangle = 0$$ as $$\langle Tv, Tv\rangle = \langle v, v \rangle$$ using linearity and the definition of the adjoint.
Expressing the vector $v$ in terms of the given orthonormal basis $e_i$ we see that \begin{eqnarray*} \langle Tv, Tv\rangle &=& \langle T\sum v_i e_i, T\sum v_j e_j\rangle \\ &=& \langle \sum v_i T(e_i), \sum v_j T(e_j)\rangle \\ &=& \sum v_i\sum \overline{v_j}\langle T(e_i), T(e_j)\rangle \\ &=& \sum v_i\sum \overline{v_j}\delta_{ij} \\ &=& \langle v, v \rangle \\ \end{eqnarray*}