The Kreps–Yan theorem for Banach ideal spaces

Consider a closed convex cone $C$ in a Banach ideal space $X$ on some measure space with $\sigma$-finite measure. We prove that the fulfilment of the conditions $C\cap X_+=\{0\}$ and $C\supset-X_+$ guarantees the existence of a strictly positive continuous functional on $X$ whose restriction to $C$ is nonpositive.