Computing infima on convex sets, with applications in Hilbert spaces

Using intuitionistic logic, we prove that under certain reasonable conditions, the infimum of a real-valued convex function on a convex set exists. This result is then applied to problems of simultaneous approximation in Hilbert space $H$ and the corresponding operator space $\mathcal{B}\left(H\right).$ This enables us to establish that a bounded, weak-operator totally bounded, convex subset of $\mathcal{B}\left( H\right)$ is strong-operator located.