Determining continuous functions on Banach spaces

2012-04-25T20:16:13Z

Let $X$ be a real Banach space.

For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ determines $g$ if whenever $g(x) \neq g(y)$, there is an $f \in \mathcal{F}$ such that $f(x) \neq f(y)$. Since $\mathcal{F}=X^*$ determines any function (because in fact it "determines points"), it makes sense to define: $$s(g):=\min \{|\mathcal{F}|:\mathcal{F} \mbox{ determines } g\}.$$

For example if $g$ is constant then $s(g)=0$ and if $g$ is linear then $s(g)=1$.

Now let $$s(X)=\sup_{g \in C(X)} s(g).$$

For finite-dimensional $X$ we have that $s(X) = \dim(X)$ and the supremum is attained for example by the function $g(x)=\|x\|$.

I hope someone with more background in Banach spaces than me (that probably includes most of the regulars on this site!) can easily answer some/all of the following:

1) Is there a simple way to compute $s(X)$ for infinite-dimensional $X$?

2) Is there always a $g \in C(X)$ for which $s(X)=s(g)$?

3) Is it true that if $X$ is separable then $s(X) \leq \aleph_0$?

Any comment, answer or reference will be greatly appreciated.

Answer by Michael Renardy for Determining continuous functions on Banach spaces

2012-04-25T21:28:31Z

If the span of ${\cal F}$ is weak-* dense, it separates points. If not, it does not determine $\|x\|$.

Determining continuous functions on Banach spaces - MathOverflow

Determining continuous functions on Banach spaces

Answer by Michael Renardy for Determining continuous functions on Banach spaces