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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.

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  • $\begingroup$ Think about an open angle or cone.. $\endgroup$
    – Berci
    Nov 21 '18 at 11:23
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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].

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

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