# Explicit formula for inner product given orthonormal basis

Suppose $$V$$ is a $$\mathbb C$$-vector space and $$v_1, v_2, \ldots v_n \in V$$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $$\langle\cdot{,}\cdot\rangle$$. Is there any algorithm for explicit formula for this inner product?

Using the sesquilinearity of the inner product, we have $$\left \langle \sum_{j=1}^n \alpha_j v_j , \sum_{k=1}^n \beta_K v_k \right \rangle = \sum_{j=1}^n \sum_{k=1}^n \alpha_j \overline{\beta_k} \langle v_j,v_k \rangle = \sum_{j=1}^n \alpha_j \overline{\beta_j}$$ That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".