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.^[1] The result can be applied in probability theory when dealing with paths of the brownian motion.^[2]

Let ${\displaystyle [0,T]}$ be an intervall and let ${\displaystyle \mathbb {T} =(\tau _{n})_{n\in \mathbb {N} }}$ be a sequence of dyadic partitions of ${\displaystyle [0,T]}$.

Let ${\displaystyle C^{\alpha }([0,T],\mathbb {R} )}$ for ${\displaystyle 0<\alpha <1}$ be a Banach space of Hölder continuous functions with norm

${\displaystyle \|f\|_{C^{\alpha }}=\|f\|_{\infty }+|f|_{C^{\alpha }}:=\sup \limits _{t\in [0,T]}|f(t)|+\sup \limits _{\begin{matrix}s,t\in [0,T]\\s\neq t\end{matrix}}{\frac {|f(s)-f(t)|}{|s-t|^{\alpha }}}}$

and ${\displaystyle \ell ^{\infty }(\mathbb {R} )}$ be the Banach space of bounded sequence with supremum norm

${\displaystyle \|a\|_{\infty }:=\sup \limits _{n}|a_{n}|}$.

The map ${\displaystyle S\colon C^{\alpha }([0,T],\mathbb {R} )\to \ell ^{\infty }(\mathbb {R} )}$ defined as

${\displaystyle f\mapsto \left(2^{(m+1)\left(\alpha -{\tfrac {1}{2}}\right)}|\theta (f)_{m,k}^{\mathbb {T} }|\right)_{m,k},\qquad (m,k)\in \mathbb {N} _{0}\times \{0,1,\dots ,2^{m}-1\}}$

is an isomorphism, where ${\displaystyle \theta (f)_{m,k}^{\mathbb {T} }}$ are the Schauder coefficients of ${\displaystyle f}$ along ${\displaystyle \mathbb {T} }$ of ${\displaystyle [0,T]}$.

The Schauder coefficients are

${\displaystyle \theta (f)_{m,k}^{\mathbb {T} }=\langle f',h_{m,k}\rangle _{L^{1}}=\int _{0}^{T}f'(x)h_{m,k}(x)dx.}$

for Haar functions ${\displaystyle h_{m,k}}$ based on the dyadic partition ${\displaystyle \tau _{m}}$.

• The result was generalized in 2025 for general partitions.^[3]