Pythagorean Theorem (Hilbert Space)

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Theorem

Let $H$ be a Hilbert space with inner product norm $\left\|{\cdot}\right\|$.

Let $f_1, \ldots, f_n \in H$ be pairwise orthogonal.


Then:

$\displaystyle \left\|{\sum_{i \mathop = 1}^n f_i}\right\|^2 = \sum_{i \mathop = 1}^n \left\|{f_i}\right\|^2$


Proof

\(\displaystyle \left\Vert{\sum_{i \mathop = 1}^n f_i}\right\Vert^2\) \(=\) \(\displaystyle \left\langle{\sum_{i \mathop = 1}^n f_i, \sum_{j \mathop = 1}^n f_j}\right\rangle\) $\quad$ Definition of Inner Product Norm $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \left\langle{f_i, f_j}\right\rangle\) $\quad$ Linearity of Inner Product $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \begin{cases}\left\langle{f_i, f_i}\right\rangle & i = j\\ 0 & i \ne j\end{cases}\) $\quad$ The $f_i$ are pairwise orthogonal $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \left\Vert{f_i}\right\Vert^2\) $\quad$ Definition of Inner Product Norm $\quad$

$\blacksquare$


Also see

  • Pythagoras's Theorem, the well-known instance of this theorem with $H = \R^2$ and the usual inner product.


Source of Name

This entry was named for Pythagoras of Samos.


Sources