The Pythagorean Identity for Inner Product Spaces

The Pythagorean Identity for Inner Product Spaces

 Theorem 1 (The Pythagorean Identity for Inner Product Spaces): Let $H$ be an inner product space. If $x, y \in X$ are orthogonal then $\| x + y \|^2 = \| x \|^2 + \| y \|^2$.
• Proof: Since $x$ is orthogonal to $y$ we have that $\langle x, y \rangle = \langle y, x \rangle = 0$. So:
(1)
\begin{align} \quad \| x + y \|^2 &= \langle x + y, x + y \rangle \\ &= \langle x, x \rangle + \langle x, y \rangle + \langle y, x \rangle + \langle y, y \rangle \\ &= \| x \|^2 + \| y \|^2 \quad \blacksquare \end{align}
 Theorem 1 (The Generalized Pythagorean Identity for Inner Product Spaces): Let $H$ be an inner product space. If $x_1, x_2, ..., x_n \in X$ are pairwise orthogonal then $\displaystyle{\left \| \sum_{k=1}^{n} x_k \right \|^2 = \sum_{k=1}^{n} \| x_k \|^2}$.