Summation of scalarproducts of orthographic vector

Question

Let $V$ a linear space with dimension $2n$ and scalar product $< \cdot, \cdot >$, $G$ a linear subset with dimension $n$. Let $e_1, \dots e_{2n} $ a orthonormal base of $V$, $g_1, \dots, g_n$. How can we follow $ \sum_{j = 0}^{2n} <e_j, g_i>^2 = || g_i ||^2 $ for $ i \in \{0, \dots, n \} $ ?

Answer 1

Given a orthonormal base $\{e_1,\ldots,e_{2n}\}$ and any vector $v$ is true that $$v=\sum_{j=1}^{2n}\langle v,e_j\rangle e_j.$$ Thus, $$\|v\|^2=\sum_{j=1}^{2n}\langle v,e_j\rangle^2.$$