Determining continuous functions on Banach spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:27:37Z http://mathoverflow.net/feeds/question/95193 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95193/determining-continuous-functions-on-banach-spaces Determining continuous functions on Banach spaces Ramiro de la Vega 2012-04-25T20:16:13Z 2012-05-23T23:22:00Z <p>Let $X$ be a real Banach space. </p> <p>For a continuous (not necessarily linear) function $g:X \to \mathbb{R}$ and a family $\mathcal{F} \subseteq X^*$, we´ll say that $\mathcal{F}$ <em>determines</em> $g$ 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: <code>$$s(g):=\min \{|\mathcal{F}|:\mathcal{F} \mbox{ determines } g\}.$$</code></p> <p>For example if $g$ is constant then $s(g)=0$ and if $g$ is linear then $s(g)=1$. </p> <p>Now let $$s(X)=\sup_{g \in C(X)} s(g).$$</p> <p>For finite-dimensional $X$ we have that $s(X) = \dim(X)$ and the supremum is attained for example by the function $g(x)=\|x\|$.</p> <p>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:</p> <blockquote> <p>1) Is there a simple way to compute $s(X)$ for infinite-dimensional $X$?</p> <p>2) Is there always a $g \in C(X)$ for which $s(X)=s(g)$?</p> <p>3) Is it true that if $X$ is separable then $s(X) \leq \aleph_0$?</p> </blockquote> <p>Any comment, answer or reference will be greatly appreciated.</p> http://mathoverflow.net/questions/95193/determining-continuous-functions-on-banach-spaces/95199#95199 Answer by Michael Renardy for Determining continuous functions on Banach spaces Michael Renardy 2012-04-25T21:28:31Z 2012-04-25T21:28:31Z <p>If the span of ${\cal F}$ is weak-* dense, it separates points. If not, it does not determine $\|x\|$.</p>