It is well known that locally convex spaces are both characterized as vector spaces in which the topology is determined by a family of seminorms as well as topological vector spaces having a 0-neighbourhood base of (absolutely) convex sets.


Most texts on locally convex spaces heavily use (absolutely) convex subsets in the development of the theory. In fact, one could make a dictionary translating a concept about seminorms to one about absolutely convex subsets (using the Minkowski functional of the absolutely convex set), and vice versa.


Does there exist a text which develops the theory of locally convex spaces mainly using seminorms, downplaying the use of (absolutely) convex sets?


Does there exist a text which develops the above-mentioned dictionary (beyond showing the equivalence between both definitions of locally convex space)?



Locally convex spaces and seminorms
Such a text is


Garnir, Henri G.; De Wilde, Marc; Schmets, Jean:
Analyse fonctionnelle : Théorie constructive des espaces linéaires à semi-normes Tome 1, Théorie générale, Birkhäuser Verlag 1968


