Let $S$ be a closed subspace of a Hilbert space $H$, and $p_S$ be the orthogonal projection function onto $S$. I would like to know why $\forall u \in H$, $\|u\| = \|p_s(u)\|$ where $\|\cdot\|$ is induced by the scalar product of $H$. I certainly know $p_s$ is linear and continuous, but couldn't quite figure out why such equality holds. Any help will be greatly appreciated.

  • $\begingroup$ You probably mean an inequality, because an orthogonal projection in general maps some non-zero vectors to zero. In other words, the equality you wrote is not true. $\endgroup$ – uniquesolution Nov 24 '17 at 19:12
  • $\begingroup$ This is not true as stated. $\Vert p_S(u) \Vert \le \Vert u \Vert$, but for $u \in H \setminus S$ $\Vert p_S(u) \Vert < \Vert u \Vert$. For $u \in S$, $p_S(u) = u$ so $\Vert p_S(u) \Vert = \Vert u \Vert$. $\endgroup$ – Robert Lewis Nov 24 '17 at 19:13
  • $\begingroup$ @uniquesolution Thank you so much it makes sense $\endgroup$ – James Nov 24 '17 at 19:17
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    $\begingroup$ @RobertLewis Thank you for the insight, it makes sense. $\endgroup$ – James Nov 24 '17 at 19:18

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