Parallelogram Identity is Useful!

2026-01-30

I have only taken one (introductory) class on functional analysis. As such my only memory of the parallelogram identity $$ ||v+w||^2 + ||v-w||^2 = 2 \left( ||v||^2 + ||w||^2 \right) $$ is mostly as a curiosity which can be used to determine whether a Banach space is Hilbert. I just found out this is not its only use!

For an unrelated reason detailed below, I was trying to show that there is a projection to any closed subspace $V$ of a Hilbert space $H$. This seems obvious, but I had a hard time showing that fixing $h \in H$, a sequence $v_n \in V$ minimizing $\left< v_n,h \right>$ is Cauchy, thus giving $v \leftarrow v_n$ a candidate in $V$ for projection of $h$ to $V$. This stack exchange post gives a proof using the parallelogram identity! To get that $v_n$ is Cauchy, write $||v_n - v_m ||^2$ creatively add $0 = h-h$ and use the parallelogram identity — the rest is routine.

The moral of the story is that if you have a Hilbert space and you want to get a Cauchy sequence, give the parallelogram identity a shot.

Reason I care, which is likely not a reason you will care: I wanted to read a functional analysis proof of the existence and uniqueness of the conditional expectation, which is basically a nice projection from $L^1$ of a sigma algebra to $L^1$ of a sub-sigma algebra. The conditional expectation is a useful gadget for some proofs in random dynamics that I’m learning in the seminar this quarter. It is basically a Radon-Nikodym derivative, but there is also a functional analysis way to prove that the existence and uniqueness of conditional expectation that I want to know. It is my impression that in the wild, functional analysis methods can be much faster than measure theory when you can pull it off. If for some reason you’re interested in this (I don’t know why you would be if you didn’t need it for some application) I’m reading out of Chapter 5 of Einseidler and Ward.