Pythagorean Theorem (Hilbert Space)

From ProofWiki
Jump to navigation Jump to search


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

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


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


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


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.