Ciesielski isomorphism

In functional analysis, the Ciesielski's isomorphism establishes an isomorphism between the Banach space of Hölder continuous functions ${\displaystyle C^{\alpha }([0,T],\mathbb {R} )}$, equipped with a norm, and the space of bounded sequences ${\displaystyle \ell ^{\infty }(\mathbb {R} )}$, equipped with the supremum norm, by coefficients of a Schauder basis along a sequence of dyadic partitions.

The statement was proved in 1960 by the Polish mathematician Zbigniew Ciesielski. The result can be applied in probability theory when dealing with paths of the brownian motion.