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2026-03-26 07:55:32.1774511732
Prove of inner product space and orthonormal system's necessary condition to be complete
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Suppose the system is complete and there exists $\;v\in Y\;\;s.t.\;\;v\perp u_i\;\;\forall\,i\;$. Then
$$v=\sum_{i\in I} a_iu_i\implies \forall\,k\;,\;\;0=\langle v,u_k\rangle=\left\langle\sum_{i\in I}a_iu_i\,,\, u_k\right\rangle=\sum_{i\in I}a_i\langle u_i,u_k\rangle=a_k$$
and from here $\;v=0\;$