The set of bounded and Riemann integrable maps are a Banach space?

Question

The set of bounded and Riemann integrable maps, with the norm $\|\cdot\|_{\infty}$ are a Banach space? To be more clear of what am I talking about, the set is $$\mathcal{RI}:= \{\phi: A\rightarrow B: \phi \text{ is bounded and Riemann integrable}\}$$ where $A$ is a box in $\mathbb{R}^d$