Dual space of continuous functions - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:12:31Z http://mathoverflow.net/feeds/question/83593 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83593/dual-space-of-continuous-functions Dual space of continuous functions Mariarty 2011-12-16T04:45:43Z 2011-12-16T04:52:37Z <p>Let $C_b(\Omega,V )=$ { $f:\Omega\rightarrow V$ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)\|$. Let $C_b(\Omega)=C_b(\Omega,\mathbb R)$. For a normal topological space $\Omega$ ( $T_4$-space) it holds that</p> <p>$$C_b(\Omega)^*=rba(\Omega),$$</p> <p>where $rba(\Omega)$ is the space of regular bounded finitely additive measures, and also $$x^*f=\int\limits_{\Omega}f(\omega)\mu(d\omega),\quad f\in C_b(\Omega),\quad x^*\in C_b(\Omega)^*,\quad \mu\in rba(\Omega)$$</p> <p>Are there more precise results for the case $C_b (\mathbb R)$? Particularly I'm interested in more "beautiful" presentation of the measure $\mu$.</p>