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**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.