The initial condition for a DDEs system is defined as Banach space. I am having a problem to understand what is Banach space and the uniform norm.
For example,
The initial condition take the form \begin{equation} S(\phi)=\psi_1(\phi),\quad I(\phi)=\psi_2(\phi),\quad P(\phi)=\psi_3(\phi), \end{equation} where $\psi=(\psi_1,\psi_2,\psi_3)^\top\in C_+$ such that that $\psi_i(\phi)\geq 0\quad (i=1,2,3), \forall \phi\in[-\tau,0]$, and $C_+$ denotes the Banach space $C_+([-\tau,0],\mathbb{R}_+^3)$ of continuous functions, mapping the interval $[-\tau,0]$ into $\mathbb{R}_+^3$ and denotes the norm of an element $\psi$ in $C_+$ by \begin{equation} \|\psi\| = \sup\limits_{-\tau\leq\phi\leq0}\{|\psi_1(\phi)|,|\psi_2(\phi)|,|\psi_3(\phi)|\}.\quad \quad (*) \end{equation}
The question: what exactly is (*) and Banach space? How would I interpret/explain this initial condition in words?