How many inner products exist in $R^n$? I want to know how many inner products exist in $R^n$ making it a Hilbert space. I know they all can be corresponded to the Euclidean inner product by some Isometry/Unitary function, but I want to know more explicit formula for this function, and if it is twice differentiable or not ?

Given an inner product $\langle \cdot, \cdot \rangle$ in $\mathbb{R}^n$, consider a basis $\{ 
v_i \}_{i=1}^n$ of $\mathbb{R}^n$ and define $g_{ij} = \langle v_i, v_j \rangle$.
Take $u=\sum_{i=1}^n \alpha_i v_i$ and $v=\sum_{j=1}^n \beta_j v_j$.
Then 
$$(1) : \langle u, v \rangle = \sum_{i, j = 1}^n g_{ij} \alpha_i \beta_j$$
The matrix $[g_{ij}]$ so defined is symmetric and positive.
Conversely, given a symmetric positive matrix $[g_{ij}]$, the bilinear form defined by $(1)$ is an inner product, as one can verify.
