Are metric vector spaces interesting? There are topological spaces, metric spaces, normed vector spaces and topological vector spaces. However, metric vector spaces are hardly mentioned anywhere. Can you provide a book, script or survey on the topic of metric vector spaces or provide a detailed explanation while such don't exist?
Search engine simply lead me to the comparison between metric spaces and vector spaces. And I know that convex sets can be quite rare in metric vector spaces.
 A: For most mathematicians, a metric on a vector space would not be interesting unless the metric topology is compatible with the vector space structure. This means that you want vector addition and scalar multiplication to be continuous operations with respect to the metric topology. Without this requirement, there's not much point to considering a vector space with a metric; the topology and the algebra must interact for a metric vector space to be a useful concept.
Normed vector spaces are, of course, an important example of this, but there exist vector spaces with a compatible metric topology which are considered interesting. The most notable example are Fréchet spaces, a specific type of locally convex space.
A locally convex space is a topological vector space whose topology is generated by a family of seminorms $(\rho_\alpha)_{\alpha\in I}$. A Fréchet space is a locally convex space which is completely metrizable via a translation-invariant metric. These are not Banach spaces; they are strict generalizations of normed vector spaces, because a translation-invariant metric does not necessarily define a norm. (However, every Banach space is trivially a Fréchet space.) In most practical cases, when one has a Fréchet space it is because one has access to a countable family of seminorms, say $(\rho_n)_{n=0}^\infty$, which generates the topology and separates points (meaning $x = 0$ if and only if $\rho_n(x) = 0$ for all $n$). When one has a countable family of seminorms like so, then one can define a complete translation-invariant metric on the space by
$$
d(x,y) = \sum_{n=0}^{\infty} 2^{-n}\frac{\rho_n(x-y)}{1+\rho_n(x-y)}.
$$
The reason Fréchet spaces are important is that they are the foundational objects in the theory of distributions/generalized functions, widely known as the area of analysis that puts objects like the Dirac delta on rigorous footing. In particular, to every topological vector space you can associate its topological dual, the space of continuous linear functions on the TVS. When you take the topological dual of a Fréchet space, you get the space of distributions, where objects like the Dirac delta live. Fréchet spaces are also fairly concrete: a simple and standard example is Schwartz space $\mathcal{S}(\mathbb{R}^d)$, the space of function $f:\mathbb{R}^d\to\mathbb{C}$ satisfying
$$
\rho_{\alpha,\beta}(f) = \sup_{x\in\mathbb{R}^d} |x^\alpha\nabla^\beta f(x)|<\infty
$$
for all multi-indices $\alpha,\beta$. The corresponding dual space is $\mathcal{S}'(\mathbb{R}^d)$, the space of tempered distributions.
