So $V$ is an inner product space (finite dimensional) with inner product defined.
If $v$ and $w$ are vectors in $V$, how would one go about proving this?
$\langle \phi_\beta (x), \phi_\beta (y) \rangle ' = \langle [x]_\beta, [y]_\beta \rangle ' = \langle x, y \rangle$
The $\langle \rangle '$ is the standard inner product on $F^n$. $\beta$ is an orthonormal basis for V.
Attempt: It seemed to me that the first half of the equality would be obvious, because $\phi_\beta (x) =[x]_\beta$ and $\phi_\beta (y) =[y]_\beta$, or at least that is how I know $\phi_\beta$ to be defined. I don't know if that's rigorous though.
Probably this is what you are looking for. If $ \beta = \{e_1,e_2,...,e_n\} $ then for all $ v \in V $, you have $ v = \sum_i \langle v,e_i\rangle e_i$ and $ [v]_\beta = (\langle v,e_1\rangle,\langle v,e_2\rangle,.....,\langle v,e_n\rangle) \in \mathbb{F}^n $ Hence immediately using $ \langle e_i,e_j\rangle = \delta_{ij} $ you obtain $$ \langle x,y\rangle = \sum_{i=1}^n \langle x,e_i\rangle\langle y,e_i\rangle = \langle [x]_\beta,[y]_\beta\rangle' $$