Let $X=\{0,1\}^{\mathbb{N}}$ be the sequence space and $T:X\to X$ the left shift mapping. Define the vector space $\text{Bow}(X,T)$ as $$ \text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in \mathbb{N}}\sup_{d_n(x,y)\leq 1/2} |S_nf(x)-S_nf(y)|<\infty\}, $$ where

$S_nf=\sum_{j=0}^{n-1}f\circ T^j$;

$d_n(x,y)=\max\{d(T^jx,T^jy),~j=0,\ldots,n-1 \} $;

- $d(x,y)=2^{-N}$, where $N=\inf\{i\in \mathbb{N}\;; x_i\neq y_i\}$.

Consider in the $\text{Bow}(X,T)$ space the norm $$ |f|_{\text{Bow}} = 2\,\|f\|_{\infty} +\sup_{n\geq 1}\ \max_{d_n(x,y)\leq 1/2}|\,S_nf(x)-S_nf(y)\,|. $$

Question:Is $(\text{Bow}(X,T),|\cdot|_{\text{Bow}} )$ a Banach Space?