Parallelogram law

Statement

Let $V$ be an inner product space over $\R$ or $\C$, with the induced norm $\norm{\cdot}$. Then, for all $x, y \in V$, $$\norm{x + y}^2 + \norm{x - y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2.$$

Remarks

The proof expands both squares with the inner product; the mixed terms cancel. By the Jordan–von Neumann theorem, a norm comes from an inner product if and only if it satisfies this law. The norm $\abs{x_1} + \abs{x_2}$ on $\R^2$ fails it for $x = (1, 0)$ and $y = (0, 1)$, so it comes from no inner product.

Explanations

L1 human

In a parallelogram, the squares of the two diagonals add up to the squares of the four sides. The vectors $x + y$ and $x - y$ are the diagonals of the parallelogram with sides $x$ and $y$. In a rectangle, both diagonals have the same length, and this is the Pythagorean theorem twice.