Hilbert Spaces 5 | Proof of Jordan-von Neumann Theorem [dark version]

A normed space becomes a genuine Hilbert space exactly when its norm obeys the parallelogram law. That criterion—Jordan–von Neumann’s theorem—turns a geometric identity about lengths into the algebraic structure of an inner product. The proof strategy is to start with the norm and *define* a candidate inner product from it, then verify that this candidate satisfies the inner-product axioms.

The parallelogram law says that for every pair of vectors X and Y, the quantity ‖X+Y‖² + ‖X−Y‖² is always the same as 2‖X‖² + 2‖Y‖². If this identity holds universally, then one can reconstruct an inner product whose induced norm matches the original norm: taking ⟨X,X⟩ and square-rooting must return ‖X‖. In the real case, the polarization identity provides the only plausible formula for such an inner product, so the remaining task is not guessing—it’s proving that the formula actually works.

The construction uses the norm to define ⟨W,Z⟩ via a polarization-type expression (in the real setting, no complex conjugation is needed). Three inner-product properties must be checked: positive definiteness, symmetry, and linearity in the second argument. Positive definiteness is straightforward: plugging the same vector into both slots makes the formula collapse to ‖W‖² (up to the correct factor), so ⟨W,W⟩=0 forces W to be the zero vector. Symmetry is equally direct in the real case because swapping the two arguments does not change the norm-based expression.

Linearity is where the parallelogram law earns its keep. To prove linearity in the second argument, the proof repeatedly rewrites differences of norm-squares into sums so the parallelogram law can be applied. For example, to handle expressions involving ‖W+Z‖²−‖W‖², the argument artificially adds and subtracts the missing term ‖W‖² so each piece can be matched to the parallelogram pattern. Solving small “systems” of vector substitutions (effectively choosing X and Y so that X±Y becomes the needed combinations) lets the parallelogram law be used twice, after which cancellation removes the extra terms.

This yields a key scaling rule: factors of 1/2 can be pulled out of the second argument, i.e., ⟨W, (1/2)Z⟩ relates cleanly to ⟨W,Z⟩. By iterating the same reasoning, the proof extends this to factors (1/2)ⁿ for natural n. Next comes additivity in the second argument: ⟨W, Z+Ẑ⟩ becomes ⟨W,Z⟩+⟨W,Ẑ⟩ through another carefully engineered decomposition that again turns the needed expression into a form where the parallelogram law applies, with cancellation leaving exactly the desired sum.

From additivity, homogeneity for integers follows (pulling out 2, then any natural number K). The remaining real scalars are handled by approximation using rationals of the form K/2ⁿ, and then extending to all real numbers via continuity of the norm. With linearity established, the constructed inner product is valid, and the Jordan–von Neumann theorem is complete: a normed space is an inner product space if and only if the parallelogram law holds. The complex case follows the same blueprint with additional terms in the polarization identity.

In short: the parallelogram law is not just a curiosity—it’s the exact algebraic fingerprint that lets length data be upgraded into an inner product, which is why Hilbert spaces can be characterized purely by norm geometry.