My professor told us there is a theorem which states that given a finite dimensional vector space $V$ with inner product $<|>_V$, we can have some sort of relationship between the inner product $<|>_V$ and the standard inner product of $\mathbb R^n$. What is this theorem? Where can I find this theorem?
For example, $V$ is $\mathbb R_1[t]$, first order polynomials of the variable $t\in \mathbb R$, with the standard basis. Suppose we have a function from $\mathbb R_1[t] \times\mathbb R_1[t] \to \mathbb R$ defined by $$<a_0+ a_1t \mid b_0+b_2 t> = a_0b_0+a_1b_1.$$ How to show this is an inner product by showing that this function is the "same" as the standard inner product in $\mathbb R^2$?