Definition:Real-Valued Periodic Function Space

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Let $I = \closedint 0 T$ be a closed real interval.

Let $\map C I$ be the space of real-valued functions, continuous on $I$.

Let the elements of $\map C I$ be real periodic functions with the period $T$:

$f \in \map C I: \map f T = \map f 0$

Then the set of all such mappings $f$ is known as real-valued periodic (over $I$) function space and is denoted by $\map {C_{per}} I$:

$\map {C_{per}} I:= C_{per} \paren {I, \R} = \set {f : I \to \R : \map f T = \map f 0}$