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I have got a question about the topology we can get from out a separeting family of seminorms (to make a topological vector space from out an arbitrary vector space). There exists the following statement:

Let $X$ be a vectorspace and $f:X\rightarrow\mathbb{F}$ (with $\mathbb{F}=\Bbb{C}$ of $\Bbb{R}$) a linear functional. Suppose $\mathcal{F}$ is a separating family of seminorms on $X$ and introduce the locally convex topology $\tau$ on $X$ (read W.Rudin or something else). Then the following are equivalent:

  • $f$ is continuous when $X$ has the $\tau$-topology
  • There exists $n\geq1$ and $p_i\in\mathcal{F}$ and $c_i>0$ ,$i\in\{1,2,\cdots,n\}$ such that $|f(x)|\leq\sum_{i=1}^n{c_ip_i(x)}$ for all $x\in X$

How can we bring the seminorms and the topology together? I know that the topology comes from it but the other way around?! Can someone help me with this equivalences

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  • $\begingroup$ I think you may look at it from the wrong angle. We have a (Hausdorff locally convex) topological vector space $X$, and a family of seminorms that happens to generate the topology. Then a linear functional $f\colon X \to \mathbb{K}$ is continuous if and only if it is dominated by a finite number of the seminorms from the family. $\endgroup$
    – Daniel Fischer
    Sep 17, 2013 at 18:12
  • $\begingroup$ Yes, thats the statement i have to prove or not? $\endgroup$
    – codim
    Sep 17, 2013 at 18:13
  • $\begingroup$ Yes. But if proving that is your problem, I don't understand the last paragraph at all. $\endgroup$
    – Daniel Fischer
    Sep 17, 2013 at 18:15
  • $\begingroup$ Maybe i hav to delete the last paragraph. The statement is that i have no idea how to begin or to prove it -.- $\endgroup$
    – codim
    Sep 17, 2013 at 18:16
  • $\begingroup$ A linear functional $f \colon X \to \mathbb{K}$ is continuous if and only if $\{ x \in X : \lvert f(x)\rvert < 1\}$ is a neighbourhood of $0$ in $X$. The rest is just the definition of the topology defined by the family of seminorms. $\endgroup$
    – Daniel Fischer
    Sep 17, 2013 at 18:19

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