# Explicit norm on \mathcal{C}^0(\mathbb{R},\mathbb{R})\mathcal{C}^0(\mathbb{R},\mathbb{R})

Do you know an explicit norm on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$? Using the axiom of choice, every vector space admits a norm but have you an explicit formula on $\mathcal{C}^0(\mathbb{R},\mathbb{R})$?

A related question is: Can we proved that $\mathcal{C}^0(\mathbb{R},\mathbb{R})$ has a norm without the axiom of choice?

The answer is no. There is no explicit norm on $\mathcal{C}^0(\mathbb{R}, \mathbb{R})$; constructing any norm on this space requires the axiom of choice to be used in an essential way.
In my answer to the (newer) question Inner product on $C(\mathbb R)$, I show that it is consistent with ZF+DC that there does not exist a norm on the vector space $\mathcal{C}^0(\mathbb{R}, \mathbb{R})$ (called $C(\mathbb{R})$ in that question).