# A closed convex set in a complex Banach space.

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.

• Think about an open angle or cone.. Nov 21 '18 at 11:23

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].

• But K in my question is not a subspace. Nov 21 '18 at 13:56
• @Infinity Any linear subspace satisfies your conditions on $K$. Nov 21 '18 at 23:09