If $x_n \to x$ weakly then $\exists y_n \in \operatorname{co}\{x_1, \dots, x_n\}$ s.t. $y_n \to x$.

Question

Let $X$ be a normed space and $x_n \to x$ weakly. I can show that there exists $\{y_n\} \subseteq \operatorname{co}\{x_1, x_2, \dots \}$ s.t. $\|y_n -x \| \to 0$ where co is convex hull. To show the existence of sequence, I used the following theorem

1.4. Theorem. If $\mathscr{X}$ is $a$ LCS and $A$ is a convex subset of $\mathscr{X}$, then $\mathrm{cl} A=\mathrm{wk}-\mathrm{cl} A$.

But I used $x\in \operatorname{wk-cl}(\{x_1, x_2, \dots \})\subseteq \operatorname{wk-cl}(\operatorname{co}\{x_1, x_2, \dots \})=\operatorname{cl}(\operatorname{co}\{x_1, x_2, \dots\})$ but I can't show $y_n \in \operatorname{co}\{x_1, x_2, \dots x_n\}$. I know that there exists a sequence, perhaps I need to show that each $y_n$ lies in the convex hull of finite number of $x_n$'s. Please give me some ideas.

Answer 1

Suppose $y_n \in co \{x_1,x_2,...,x_{k_n}\}$ with $k_1<k_2<....$ Consider a sequence of the type $\{x_1,x_1,...,x_1, y_1,y_1,...,y_{1},y_2,y_2,...,....$ where you have blocks of size $k_1-1,k_2-1,...$. This satisfies your requirements.

Thanks to David Mitra for the suggestion.