# Are There "Unnormed" Vector Spaces?

by zooxanthellae
Tags: spaces, unnormed, vector
 P: 159 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.
 P: 366 I'm not an expert on this sort of thing, but a norm always induces a metric. So if we find an unmetrizable vector space it is a fortiori "unnormable." Some examples can be found here.
 Sci Advisor HW Helper P: 9,421 infinite dimensional vector spaces may have many different topologies of interest, not all defined by norms. so even if a vector space can have a norm, the topology relevant to certain problems may not be a normed topology. in this context it seems the previous poster meant to say there are topological vector spaces whose topology is not definable by a norm.
P: 1,168

## Are There "Unnormed" Vector Spaces?

zooxanthellae wrote:

" 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. "

In another sense, if your (bare-bones) vector space V is finite-dimensional, you can always use the isomorphism with ℝ n to pullback the norm. But there are other issues, as Mathwonk said: if you have a topology given by a metric , the topology may not be generated by a norm; there are specific conditions under which a given metric is generated by a norm, i.e., so that there is a norm ||.|| with d(x,y):=||x-y|| ; I think one of the conditions is that the distance function is translation-invariant. Is that your question?
 P: 367 There is a concept of a topological vector space. This is a vector space where scalar multiplication and vector addition are continuous. Any normed vector space is a topological vector space. But there are Topological vector spaces whose topology is not given by a norm. These guys are an important example: http://en.wikipedia.org/wiki/Fr%C3%A9chet_space
 Sci Advisor P: 1,168 Maybe something else to add, re what I understood your original question to be, is that you can give your vector space more, or less structure; an unnormed vector space would be one in which you only consider the linear-algebraic aspects of the space: bases, linear independence, transformations (automorphisms), etc. This may be what is meant by an unnormed space, a space in which you only consider linear-algebraic properties and ignore others; it is like having a map of the world in which you point out some aspects but not others, like political maps or topographical ones, etc. Maybe another important property to consider is that any two norms on a finite-dimensional space generate the same/equivalent topologies.
 P: 192 The important point is that a normed vector space is not one where we can define a norm, it's one where we defined a particular one. A single vector space could hace many different norms, and it's not a normed space until we've specified one. Different norms lead to very different properties of the space, and sometimes we're just not interested in the properties that a norm gives us. That's when we have a "unnormed" space. In short, every space can be unnormed, if we choose to ignore its norm. It's only normed if we've specified one though, regardless of whether or not it is "normable."