Let $H$ and $K$ be compact subgroups of locally compact group $G$. By considering the double coset space $K\setminus G/H$, which equipped with an $N$-strongly quasi invariant measure $\mu$, for $1\leq p\leq +\infty$, we make a norm decreasing linear map from $L^p(G)$ onto $L^p(K\setminus G/H,\mu)$ and demonstrate that it may be identified with a quotient space of $L^p(G)$. In addition, we illustrate that $L^p(K\setminus G/H, \mu)$ is isometrically isomorphic to a closed subspace of $L^p(G)$. These assist us to study the structure of the classical Banach space created on a double coset space by those produced on topological space. تفاصيل المقالة