We give a characterization of the Korovkin subspaces of the space $\mathcal{R}(X)$ of Riemann integrable functions over a Hausdorff compact topological space $X$, equipped with the $\mathcal{R}$-sequential convergence. Some applications are presented in the context of the space of $2\pi$-periodic real functions which are Riemann integrable on the compact real interval $[0,2\pi]$ and of the space of Riemann integrable functions on the standard simplex and the hypercube of $\mathbb{R}^p(p\geq 1)$.
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