Parallelogram Identity is Useful!

2026-01-30

My first memory of the parallelogram identity (from either a linear algebra or analysis class) $$ ||v+w||^2 + ||v-w||^2 = 2 \left( ||v||^2 + ||w||^2 \right) $$ is mostly as a curiosity which can be used to determine whether a Banach space is Hilbert. This is not its only use!

For an unrelated reason, 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 $0 = h-h$
• conclude using 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.