In functional analysis, the Ciesielski's isomorphism establishes an isomorphism between the Banach space of Hölder continuous functions , equipped with a norm, and the space of bounded sequences , 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.
Ciesielski's isomorphism Let be an intervall and let be a sequence of dyadic partitions of .
Let for be a Banach space of Hölder continuous functions with norm
and be the Banach space of bounded sequence with supremum norm .
The map defined as
is an isomorphism, where are the Schauder coefficients of along of .
The Schauder coefficients are
for Haar functions based on the dyadic partition .
Properties The result was generalized in 2025 for general partitions.
References