When is $\| f \|_\infty$ a norm of the vector space of all continuous functions on subset S? Let S be any subset of $\mathbb{R^n}$. Let $C_b(S)$ denote the vector space of all bounded continuous
functions on S. For $f \in C(S)$, define $\| f \|_\infty = \sup_{x \in S} |f(x)|$
When is this a norm of the vector space of all continuous functions on S?

For X a locally compact Hausdorff space such as an open or closed subset or non-empty intersection of open and closed subsets of R^n, the space of bounded continuous function on X with the sup norm, which is the same as the infinity norm or essential sup norm is complete.  The sup or the essential sup is always a norm on this pace of all bounded continuous function. The completion of the space of all continuous function on X with compact support with respect to the sup norm is the space of all continuous function on X which vanishes at infinity.   
