it follows that $ \{x_n\}$ is a Cauchy sequence. As $X$ is complete, then $ \lim x_n = \hat{x} $ and additionally $  \bigcap_{n=0}^\infty X_n = \hat{x}$. Thereby $\hat{x}$ is the unique fixed point of $T$ in $X_0=X$ and the unique zero of $ f$ in $X$.

Functions and Domains {#sec:FunctionDomain}
---------------------

As in [@BergJaquette; @lessard2010recent], we convert Wright’s equation into a functional equation on the space of Fourier coefficients. For a continuous periodic function $y:{\mathbb{R}}\to {\mathbb{R}}$ with frequency $\omega >0$, we may write it as: $$y(t)  = \sum_{k \in {\mathbb{Z}}} c_{k} e^{ i \omega k t }
\label{eq:FourierEquation}$$ where $ c_k \in {\mathbb{C}}$ and $\sum_{k\in {\mathbb{Z}}} |c_k|^2 < \infty$. By [@BergJaquette] it suffices to work with sequences $ \{ c_k \}_{k=1}^{\infty}$ to study periodic solutions to . This is because real-valued functions have Fourier coefficients satisfying $ c_{-k} =c^*_k$, and periodic solutions to necessarily satisfy $c_0 =0$. Hence we define the following Banach spaces: $$\begin{aligned}
\ell^1 :=& \left\{ \{ c_k \}_{k=1}^\infty : c_k \in {\mathbb{C}}\mbox{ and } \sum_{k=1}^\infty | c_k| < \infty  \right\} & \| c \|_{\ell^1} =& 2 \sum_{k = 1}^\infty | c_k| \\
\Omega^s :=& \left\{ \{ c_k \}_{k=1}^\infty : c_k \in {\mathbb{C}}\mbox{ and } \sup_{k \in{\mathbb{N}}} 