Definition:Real-Valued Periodic Function Space

Definition
Let $\CC \closedint 0 T$ be the space of continuous on closed interval real-valued functions.

Let the elements of $\CC \closedint 0 T$ be real periodic functions with the period $T$:


 * $f \in \CC \closedint 0 T: \map f T = \map f 0$

Then the set of all such mappings $f$ is known as real-valued periodic function space and is denoted by $\CC_{per} \closedint 0 T$:


 * $\CC_{per} \closedint 0 T:= \CC_{per} \paren {\closedint 0 T, \R} = \set {f : \closedint 0 T \to \R : \map f T = \map f 0}$