  In this paper we prove that the space $ \mathbb{P}_p $ of all periodic
function of fundamental period $ p $ is a direct sum of the space $
\mathbb{P}_{p/2} $ of periodic functions of fundamental period $ p/2 $ and the
space $ \mathbb{AP}_{p/2} $ of antiperiodic functions of fundamental anti
period $ p/2 $. The decomposition can be continued by applying the
decomposition process to the successively raising periodic subspaces. It is
shown that, under certain condition, a periodic function can be written as a
convergent infinite series of anti periodic functions of distinct fundamental
anti periods. In addition, we characterize the space of all periodic functions
of period $ p \in \mathbb{N} $ in terms of all its periodic and antiperiodic
subspaces of integer periods (or anti periods). We show that the elements of a
subspace of such a space of periodic functions take a specific form (not
arbitrary) of linear combinations of the shifts of the elements of the given
space. Lastly, we introduce a lattice diagram named-periodicity diagram for a
space of periodic function of a fixed period $ p \in \mathbb{N} $. As a
particular example, the periodicity diagram of $ \mathbb{P}_{12} $ is shown.
