F-space

In functional analysis, an F-space is a vector space V over the real or complex numbers together with a metric d : V × V → R so that

Some authors call these spaces Fréchet spaces, but usually the term is reserved for locally convex F-spaces. The metric may or may not necessarily be part of the structure on an F-space; many authors only require that such a space be metrizable in a manner that satisfies the above properties.

Clearly, all Banach spaces and Fréchet spaces are F-spaces. In particular, a Banach space is an F-space with an additional requirement that d(αx, 0) = |α|⋅d(x, 0).

The L^p spaces are F-spaces for all p ≥ 0 and for p ≥ 1 they are locally convex and thus Fréchet spaces and even Banach spaces.

${\displaystyle \scriptstyle L^{\frac {1}{2}}[0,\,1]}$ is an F-space. It admits no continuous seminorms and no continuous linear functionals — it has trivial dual space.

Let ${\displaystyle \scriptstyle W_{p}(\mathbb {D} )}$ be the space of all complex valued Taylor series

...
Wikipedia