Topology determined by a countable collection of norms It is known that a locally convex space is metrizable if and only if its topology is generated by a countable separating collection of seminorms. Do those metrizable locally convex spaces that are generated by a countable collection of norms have a special distinguished interest in functional analysis?
 A: Note that a metrizable LCS is generated by a sequence of (proper) norms if it admits a (single) continuous norm. This is sometimes useful. For example,
a classical theorem of Borel states that the operator $C^\infty([-1,1]) \to \mathbb C^{\mathbb N_0}$, $f\mapsto (f^{(n)}(0))_{n\in\mathbb N_0}$ is surjective and one may ask, whether there is a continuous linear operator
$R:\mathbb C^{\mathbb N_0} \to C^\infty([-1,1])$ which gives for every scalar sequence $y$ such a function $R(y)\in C^\infty([-1,1])$ with the prescribed Taylor coefficients. 
Now it comes: The existence of such an extension operator would imply that the product $\mathbb C^{\mathbb N_0}$ had a continuous norm which is not true.
Sometimes the existence of of a continuous norm is helpful because then the dual of your space has a weak$^*$ bounded sets with dense linear span (namely the polar of the unit ball of the norm).
A last point: There is a rather misleading notion of countably normed Frechet spaces which are countable intersections of Banach spaces $X_{n+1} \subseteq X_n$ with continuous inclusions. Clearly, every such space has a continuous norm (e.g. the one of $X_1$) but having a continuous norm is not sufficient. This is a rather subtle difference which comes from the phenomenon that if you have a continuous injection between two normed spaces the extension to the completions need not remain injective.
This notion of countably normed Frechet spaces has some relevance for the so-called splitting theory for Frechet spaces. For example it is helpful to see that there are closed subspaces of nuclear Frechet spaces which are not complemented.
