interior of the set of positive operators

Question

Let $S$ be the set of all positive operators on $H$,$S$ is a closed cone in $B(H)$,the topology is induced by $\|\cdot\|$,is the interior $int(S)$ of $S$ empty?Suppose not,given a positive operator $T$,how to find $\delta > 0 $such that for all $U$ satisfying $\|U-T\|< \delta$,we have $U\in S$?

Answer 1

For any positive operator $T\in B(H)$ and $\varepsilon>0$, $T-\varepsilon iI$ is not positive. Thus $S$ has empty interior, and the same argument shows that the collection of self-adjoint elements is not open.