Complete metric linear spaces are F-spaces and if locally-convex then we call them Freche't spaces. Much is know about Freche't spaces and fundamentals can be found in many texts of Functional analysis like the book of Rudin. In general, "An F-space Sampler" by Kalton, Peck, and Roberts approaches the fundamentals of quasi-Banach spaces, i.e. those quasi-Banach spaces which are p-convex for some 0

Thanks a lot by your answer Geoff Diestel . I will read about Frechet's spaces, because I didn't ear about it, just now.

It is wellknown that every normed space is metric space but not vise versa.

there is a metric (mentioned in Rudin) which is not compatible with a norm.

Then there is some metric space which is not normed space.