Interior of a solid self dual cone

Question

Let $X$ be a real Hilbert space with a solid self dual positive cone $K$, that is, $\mathrm{int}(K)$ is non-empty and $K^{*}=K$. If $X$ is finite dimensional, I know that the $\mathrm{int}(K)$ = $\left\{x\in X | \langle x, y \rangle > 0 \mbox{ for all } y\in K\setminus\{0\}\right\}.$ But I am not sure whether it is true for infinite dimensional space. Can anyone please clarify it!