Fix a compact Hausdorff topological space $K$. Define $b\mathcal{B}(K)$ as the space of complex valued bounded Borel measurable functions on $K$ and define $C(K)$ as the space of complex valued continuous functions on $K$. I'm looking for a topology $\tau$ on $b\mathcal{B}(K)$ that satisfies the following condition:
- $\tau$ induces bounded pointwise convergence, i.e. a net $(f_\alpha)_{\alpha\in I}$ converges to $f$ iff $f_\alpha\rightarrow f$ pointwise and there exists $C>0$ and $\alpha_0\in I$ such that $\forall \alpha\ge\alpha_0, |f_\alpha|\le C;$
- $(b\mathcal{B}(K),\tau)$ is a complete topological vector space;
- $C(K)$ is dense in $(b\mathcal{B}(K),\tau)$.
Does there exist anything that do this job?
The problem pops up in trying to establish some kind of uniqueness by continuity (with respect to some "natural" topology $\tau$) of the extension to the whole $b\mathcal{B}(\sigma(A))$ of a homomorphism from $C(\sigma(A))$ to $\mathcal{L}(H)$ (the inverse of the Gelfand-Naimark isomorphism), where $H$ is a Hilbert space, $\mathcal{L}(H)$ is the space of bounded linear operators from $H$ to $H$, $A\in \mathcal{L}(H)$ is a normal operator, $\sigma(A)$ is the spectrum of $A$ and the mentioned extension is performed via spectral measures.