The loop space $Ω_X$ of a pointed topological space $X$ is the space of based maps from the circle $\mathbb S^1$ to $X$ with the compact-open topology.
The loop space $Ω_X$ of a pointed topological space $X$ is the space of based maps from the circle $\mathbb S^1$ to $X$ with the compact-open topology.
The loop space is equivalently a mapping space from the circle to some pointed topological space $X$: a space of loops in $X$. Here $X$ and the loops might be equipped with further geometric structure such as smooth structure and then one may consider a smooth loop space, etc.