Danish physicist Niels Bohr is not an unknown scientist to a typical passionate high school physics student. However, until my graduate school studies, I didn’t learn much about the work of his brother Harald Bohr, who was a mathematician. Both did pioneering work in sciences but Niels won the Nobel. Both were passionate footballers but Harald played in Olympics. Their father Christian – a professor of physiology – remarked that Harald was brilliant but Niels was special.

I learned about Harald’s work in my functional analysis class. By Zorn’s lemma, every Hilbert space admits an orthonormal basis (ONB). However, only separable Hilbert spaces have countable ONB. The ONB for inseparable Hilbert spaces can be constructed by following the principle of transfinite induction with the invocation of the Axiom of Choice/Zorn’s Lemma. One of the examples of ONBs of inseparable Hilbert space is the (Harald) Bohr basis – the almost periodic functions.