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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?

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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".

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