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?

**Answer**

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).

**Attribution***Source : Link , Question Author : Seirios , Answer Author : Community*