Pythagoras's Theorem (Hilbert Space)

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Theorem

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

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


Then:

$\displaystyle \norm {\sum_{i \mathop = 1}^n f_i}^2 = \sum_{i \mathop = 1}^n \norm {f_i}^2$


Proof

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

$\blacksquare$


Also see


Source of Name

This entry was named for Pythagoras of Samos.


Sources