Definition of Frechet space
Mia Lopez 
I have a question regarding the two equivalent definitions of a Frechet space (cf. Wikipedia):
According to Def.1, a Frechet space is a topological VS $X$, such that
- $X$ is locally convex ($0\in X$ has a local base of absorbent and absolutely convex sets)
- the topology of $X$ can be generated by a translation invariant metric (which is not unique).
- completeness (we don't care about that here)
According to Def.2, a Frechet space is a TVS $X$, such that
- the topology can be induced by a countable family of seminorms $\{\|\cdot\|_k\}_{k\in \mathbb N}$
- $X$ is Hausdorff
- completeness w.r.t. the family of seminorms
Now I'm interested, how those two definitions are related.
I understand the following steps:
1.2 $\Rightarrow$ 2.2 (a metric space is always Hausdorff)
1.1 $\Leftarrow$ 2.1 (the existence of a family of seminorms generating the topology is an equivalent definition to local convexity)
1.2 $\Leftarrow$ 2.1 and 2.2 (2.1 yields a pseudometric by setting $d(x,y) = \sum_{k=1}^\infty 2^{-k}\frac{\|x-y\|_k}{1+\|x-y\|_k}$ , 2.2 makes it a proper metric)
Now what I don't see is how you get a countable family of seminorms using only properties from definition 1. Property 1.1 yields a family of seminorms by using the Minkowski gauge on the local basis of $0$. The problem is that this yields a family, but not a countable one. I guess what I don't understand is how I get a countable family of seminorms via the metric $d$.
$\endgroup$1 Answer
$\begingroup$Say $X$ is locally convex with a translation-invariant metric $d$. Let $$B_n=\{x:d(x,0)<1/n\}.$$Since $X$ is locally convex, standard TVS this and that shows there is a balanced convex open set $C_n$ with $$0\in C_n\subset B_n.$$Now there is a seminorm $\rho_n$ such that $C_n$ is the unit ball of $\rho_n$.
Hint that's I think going to be useful in showing that the two topologies are the same: Continuity of scalar multiplication shows that for every $\lambda>0$ and every $n$ there exists $m$ so $\lambda B_m\subset C_n$.
$\endgroup$ 2
