Pythagoras's Theorem (Inner Product Space)

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Theorem

Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.

Let $\norm \cdot$ be the inner product norm on $\struct {V, \innerprod \cdot \cdot}$.

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


Then:

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


Proof

\(\ds \norm {\sum_{i \mathop = 1}^n f_i}^2\) \(=\) \(\ds \innerprod {\sum_{i \mathop = 1}^n f_i} {\sum_{j \mathop = 1}^n f_j}\) Definition of Inner Product Norm
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \innerprod {f_i} {f_j}\) since the Inner Product is linear in the first argument
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \begin {cases} \innerprod {f_i} {f_i} & i = j \\ 0 & i \ne j \end {cases}\) the $f_i$ are pairwise orthogonal
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \norm {f_i}^2\) Definition of Inner Product Norm

$\blacksquare$


Also see


Source of Name

This entry was named for Pythagoras of Samos.


Sources