Basic neighborhoods in weak topology

Question

I am trying to visualize the basic neighborhoods of the form

$V(x_0;\varepsilon,f_1,...,f_n) = \bigcap_{j=1}^n \{ x \in E : |f_j(x-x_0)|<\varepsilon \}$

where $x_0 \in E$, $\varepsilon>0$ and $f_1,...,f_n \in E'$ on the weak topology of a normed infinite-dimensional vector space $E$.

I had imagined some way "to see" these neighborhoods like open stripes bounded in a "finite number of directions", just like the vertical open stripes are the standard visualization of the basic neighborhoods in the product topology on a infinite product of topological spaces.

In this sense can I say that these basic neighborhoods are rotated stripes?

Answer 1

One can imagine that $V(x_0;f_1,\dots,f_n;\varepsilon)$ is the set of all $x\in X$ wich aproximates those functionals in $(x-x_0)$. So, the best way to imagine this set is consider the images of $f_1,\dots,f_n$ and draw it like in $\mathbb{R}^2$. Although, it is not helpful try to draw such a set.