Are There "Unnormed" Vector Spaces? Apologies if this question is barking up a ridiculous tree, but: as I understand it, a normed vector space is simply a vector space with a norm. This seems to suggest the existence of vector spaces without norms. My question is whether these are vector spaces for which no norm can be defined (and if so, what is an example of one?), or if the definition is just a way of making explicit that a given vector space has a norm that we can use.