$\Omega X$ is an H-space

The based loop space \(\Omega X\) has multiplication by concatenation: \[ (\alpha*\beta)(t)= \begin{cases} \alpha(2t),&0\le t\le\tfrac12,\\ \beta(2t-1),&\tfrac12\le t\le1. \end{cases} \] The constant loop is a two-sided unit up to the standard reparametrization homotopies. Concatenating three loops in the two possible parenthesizations differs only by a piecewise-linear reparametrization of \(I\), hence is homotopic through based loops. Thus multiplication is associative up to homotopy. Reversal of loops gives a homotopy inverse. Therefore \[ \boxed{\Omega X\text{ is an H-space}.} \]