Let $X$ be a compact hausdorff space, and put $C(X)$ for the $\mathbb{R}$-algebra of continuous maps from $X$ to $\mathbb{R}$.

For each point $x$, there is a multiplicative norm $|-|_x$ on $C(X)$, where $|f|_x = |f(x)| \in \mathbb{R}$. That is,

1. $|cf|_x = |c| \cdot |f|_x$

2. $|fg|_x = |f(x)| \cdot |g(x)|$.

3. $|f + g|_x \leq |f|_x + |g|_x$.

Let $Y$ be the set of multiplicative norms of $C(X)$ mod the equivalence relation where $|-| \sim |-|'$ when each is bounded by the other up to a constant. We have just constructed a function $\nu : X \rightarrow Y$. My questions are:

1. How might we put a topology on $Y$ using only the information of $C(X)$?

2. Is $\nu$ above a bijection? In light of the last question, is it a homeomorphism?

So in this view, norms are like points or generalized points. Interestingly, the submultiplicative norm $|-| : C(X) \rightarrow \mathbb{R}$ sending $f$ to $\text{sup}_x |f|_x$ seems like it could be the supremum of multiplicative norms in $Y$. That way, multiplicative norms are the "local" versions of submultiplicative norms, which would be "global".