Let $X$ be a Banach space over the field of complex numbers. Let $K$ be a subset in $X$ which contains all linear combinations $Z_1x+Z_2y$ for all $Z_1,Z_2\in \mathbb C$ with $Re(Z_1)\geq0$ & $Re(Z_2)\geq0$ and for all $x,y \in K$.
I want to prove or disprove that $K$ is closed convex in $X$. I have proved $K$ is convex, but I am stuck to prove or disprove that $K$ is closed.
Not every linear subspace of a Banach space is closed so the answer is NO. [ Eg. $X=l^{1}$, $K$ is the space of all sequences with only finite number of non-zero entries].