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 $f_1, \ldots, f_n \in V$ form an orthogonal set.
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
- Pythagoras's Theorem, the well-known instance of this theorem with $V = \R^2$ and the usual inner product.
Source of Name
This entry was named for Pythagoras of Samos.
Sources
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next): $\text{I}$ Hilbert Spaces: $\S 2.$ Orthogonality: $2.2$ The Pythagorean Theorem