I was reading Rudin Functional analysis I came across theorem of metrisation of topological vector space I had done theorem But Unble visualise function f geometrically
If someone give me some geometric interpretation that will be really useful for me to understand
Thanks a lot In advanced.
I’ll illustrate what happens when $X$ is the plane $\Bbb R^2$. We have that $\{V_n\}$ is a sequence of fast decreasing neighborhoods of the zero, for instance, we can put $V_n=\{x\in\Bbb R^2: \|x\|\le 3^{-n}\}$. Using the sequence $V_n$ we construct the family $\{A(r)\}$ of neighborhoods of zero, ordered by binary rational numbers in such a way that $A(r)\subset A(r’)$ for each $r<r’$. In our case each $A(r)$ is an open disc centered at the zero. Now $f(x)$ is simply the infinum of $r$ such that $A(r)$ contains $x$. So is if $s<f(x)$ then $x\not\in A(s)$ and if $s>f(x)$ then $x\in A(s)$.