Let $W$ be a subspace of a Banach space $X$. Which of the following are true.
a. W is closed then it is perfectly convex
b. W is perfectly convex then it is closed.
Definition of perfectly convex set says that, a subset A of X is called perfectly convex if for any bounded sequence $x_n$ in A and $a_n>0$, $\sum_{i=1}^{\infty}a_n=1$ we have $\sum_{n=1}^{\infty}a_nx_n\in A$.
Because W is subspace then it is convex, and there is a result says that closed convex sets are perfectly convex, by that W is perfectly convex. So a is true. For b I didn't get any idea to proving. I think it should be false. Can u some one help me to proceed further.
B. is false. All you need to do is find a counterexample.
A perfectly convex set need not be closed. Simply examine an open ball of radius $\epsilon$ around $\{0\}$ (in a Banach space in this example), i.e. let $A := B_{\epsilon}(0)$.