weak$^∗$ neighborhood of $x$ in $\ell_1$

Question

I have this problem Let $x \in \ell_1$ and $\epsilon>0.$ Choose an $N\in N$ such that $\sum\limits_{k=N}^{\infty}|x_k|<\epsilon$ I cannot understand why V is a weak$^∗$ neighborhood of $x$ in $\ell_1$ where $$V=\{y\in\ell_1: |y_k-x_k|<\epsilon \text{ for all}\ \ k<N\}.$$

Thanks for your help.

Answer 1

Define $O_k:=\{y\in \ell_1\mid |y_k-x_k|\lt\varepsilon\}$. We would be done if we manage to show that $O_k$ is a weak$^*$ open set. Recall that the weak$^*$ topology is defined on the dual space of a normed space $X$ by the family of seminorms $\lVert \varphi\rVert_x:=\langle \varphi,x\rangle_{X^*,X}$, $x\in X$. Defining $e_n$ has the unique element of $c_0$ such that $e_n(n)=1$ and $e_n(j)=0$ if $j\neq n$, we have $$O_k=\{y\mid \lVert y-x\rVert_{e_n}\lt\varepsilon\},$$ showing that $O_k$ is open for the weak$^*$ topology.