\begin{definition}[Definition:Real-Valued Periodic Function Space]
Let $I = \closedint 0 T$ be a closed real interval.
Let $\map \CC I$ be the space of real-valued functions, continuous on $I$.
Let the elements of $\map \CC I$ be real periodic functions with the period $T$: 
:$f \in \map \CC I: \map f T = \map f 0$
Then the set of all such mappings $f$ is known as '''real-valued periodic (over $I$) function space''' and is denoted by $\map {\CC_{per}} I$:
:$\map {\CC_{per}} I:= \CC_{per} \paren {I, \R} = \set {f : I \to \R : \map f T = \map f 0}$
\end{definition}