Jordan–von Neumann theorem

Remarks

The direction from 1 to 2 is the parallelogram law in an inner product space. For the converse in the real case, define $\ip{x}{y}$ by the polarization formula. Symmetry and $\ip{x}{x} = \norm{x}^2$ are immediate. The parallelogram law gives $\ip{x + x'}{y} = \ip{x}{y} + \ip{x'}{y}$, so $t \mapsto \ip{tx}{y}$ is an additive function of $t \in \R$. It is continuous, because the norm is continuous, so it equals $t \ip{x}{y}$. The norm $\abs{x_1} + \abs{x_2}$ on $\R^2$ fails the parallelogram law for $x = (1, 0)$, $y = (0, 1)$, since $4 + 4 \ne 2 + 2$, so it comes from no inner product.

Going deeper

For a student

The theorem shows that among the spaces $L^p([0, 1])$ with $1 \le p \le \infty$, only $L^2([0, 1])$ has a norm that comes from an inner product. For $p \ne 2$ the parallelogram law fails for the indicator functions of $[0, 1/2]$ and $(1/2, 1]$. It is a characterization of Hilbert spaces among Banach spaces, because completeness is not affected. The proof is a functional equation argument: the parallelogram law is the quadratic functional equation for $f(x) = \norm{x}^2$, and continuity turns additivity into linearity. The law only involves two vectors at a time, so a norm is an inner product norm as soon as its restriction to every two-dimensional subspace is.