# A locally convex space is normable, iff there exists a bounded and open set.

I want to show, that a locally hausdorff space is seminormable, if and only if there exists a bounded and open set. My proof goes as follows: Let $$B \subset V$$ be an open and bounded 0-neighborhood, $$V$$ a topological vector space and $$\mathcal{T}_{ V}$$ a topological local base. I claim that \begin{align}\label{EQ:**} B\cap \mathcal{T}_{ V} =\left\{ B\cap U\mid U\in \mathcal{T}_{ V} \right\} \end{align} is also a local base, making $$V$$ a locally bounded TVS.

proof

The inclusion $$B\cap U \subset U$$ makes $$B\cap \mathcal{T}_{ V}$$ finer then $$\mathcal{T}_{ V}$$. On the other hand, $$B\cap U$$ is open with respect to $$\mathcal{T}_{ V}$$ so both topologies are equivalent.

Having shown this, let $$B$$ be a bounded 0-neighbourhood and $$V$$ be locally convex. We can assume $$B$$ to be absolutely convex, since the convex hull, as well as the balanced convex hull stay bounded and that $$B$$ is absorbing (every 0-neighborhood is). Now setting $$\left\| v \right\| := \sup\limits_{ B'\in \mathcal{B}} \left\{ \mu _{ B'} (v)\right\}$$ should give the desired norm, where $$\mathcal{B}=\mathcal{T}_{ V}\cap B$$ and $$\mu _{ B'} (v)=\inf\limits_{}\left\{ t\mid v\in tB'\right\}$$ is the Minkowski functional. It is bounded, because of $$\mu _{ B'} (v)\leq \mu _{ B} (v)$$ for all $$B'=B\cap U$$, $$U \in \mathcal{T}_{ V}$$.

Is this proof correct?

## 1 Answer

The proposition you seek to prove is a version of a slightly more general one that is sometimes accredited to Kolmogorov. The more general proposition reads:

Proposition: A real or complex topological vector space is normable if and only if it is Hausdorff and possesses a convex bounded neighborhood of zero.

Dropping the Hausdorff assumption, the proposition remains true if "normable" is replaced by "seminormable".

Your proof seems to be in line with a standard one, except the supremum appearing in definition of the norm. Given that $$B$$ is a bounded absolutely convex neighborhood of zero, it suffices to consider the Minkowski functional $$\mu_{B}$$ (in your notation), which is a desired seminorm (or norm if the Hausdorff assumption is imposed).

• Thank you. Yes, I even realized that $\mu_{B'}(v)\leq \mu_B(v)$, but it didn't cross my mind to just use this for the definition of the seminorm (or norm in the Hausdorff case). Do you perhaps know some literture, where I can look up that Proposition? – Mahdimatika May 30 at 10:55
• Most books on topological vector space should have this proposition. One reference I myself use recently is "Topological vector spaces and their applications" by Bogachev and Smolyanov. Check theorem 1.5.7, page 40 in that book here. – YeZ May 30 at 11:11