why is test function space $\mathcal{A}$ complete

Question

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm dy<\infty\right\}$$ is a Banach space. $\mathcal{F}_p\phi$ denotes the partial Fourier transform defined by $$(\mathcal{F}_p\phi)(x,y):=\int_{\mathbb{R}^d}e^{-ip\cdot y}\phi(x,p)\;\mathrm dp,$$ while $C_0(\mathbb R^{2d})$ denotes the space of continuous functions which vanish at infinity. I don't know how to show the completeness. Does anyone have a hint or a reference?