Today we were introduced to locally convex spaces, defined thusly:
A vector space is locally convex iff it has a family of semi-norms $(p_i)$ such that $x=0$ if and only if $p_i(x)=0$ for all $i$.
The professor then said we would limit ourselves to countable families $(p_i)$ and introduced the following function:
$$d(x, y) = \sum_{n=0}^\infty \frac 1 {2^n} \frac {p_n(x-y)}{1+ p_n(x – y)}$$
He explained that this is always a metric, and called it the "natural" metric. Naturally, it didn't seem that natural to any of us. This same formula is also found on Wikipedia.
Is there some property of this metric which characterizes it? Is it for instance the only metric having some form of compatibility with the family of semi-norms $(p_i)$?
Best Answer
Local convexity is not really the property given in the question, but, rather, that $0$ (hence, every point in the vector space) has a local basis consisting of convex open sets. It is a not-completely-trivial theorem that there exists a set of semi-norms giving the topology. (That a "separating" family of seminorms, as in the question, gives a locally convex topology, is easy.)
If the collection of seminorms is countable (which is far from universally so!), then the topology is metrizable... but there is no canonical metric, as other comments and answers have mentioned. This illustrates the point that the map from "metrics" to "topologies" is many-to-one.
That is, that metric is not "natural", but the topology is natural.