Norm on the space of rapidly decreasing and continuous functions

In P.Malliavin’s book "Integration and probability" a continuous function $f$ defined on $\mathbb{R}^n$ is said to be of rapidly decrease if for all integer $m$ we have that the mapping $(1+\lVert x\rVert^2)^m f(x)$ goes to $0$ as $\lVert x\rVert$ goes to infinity. Then, a norm on this space is defined by $\lVert f\rVert_{m,0}=\max_{x\in\mathbb{R}^n}(1+\lVert x\rVert^2)^m\lvert f(x)\rvert$.

I do not understand the use of the maximum instead of the supremum. I think we may be able to prove that the above mapping is bounded and use the supremum. However I cannot see why this supremum should be reached.

If you have any idea on what’s going on here or some hints to provide, I will be happy to read it !

Thank you