Pythagorean theorem in an inner product space

Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the induced norm $\norm{\cdot}$. If $x, y \in V$ are orthogonal, then $$\norm{x + y}^2 = \norm{x}^2 + \norm{y}^2.$$

Remarks

The proof: $\norm{x + y}^2 = \norm{x}^2 + 2 \operatorname{Re} \langle x, y \rangle + \norm{y}^2$. For pairwise orthogonal $x_1, \dots, x_k$, $\norm{x_1 + \cdots + x_k}^2 = \norm{x_1}^2 + \cdots + \norm{x_k}^2$. In a real space, the converse holds; in a complex space, the equality only gives $\operatorname{Re} \langle x, y \rangle = 0$.

Explanations

L1 human

This is the theorem of Pythagoras for vectors. Two orthogonal vectors are the legs of a right triangle, and their sum is the hypotenuse. It holds in every inner product space, even in spaces of functions.