Are There Unnormed Vector Spaces?

[zooxanthellae]

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.

 

[spamiam]

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.

 

[mathwonk]

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.

 

[Bacle2]

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?

 

[deluks917]

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

 

[Bacle2]

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.

 

[alexfloo]

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."

 

[Bacle2]

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."

Still, tho, in a finite-dimensional space, all norms are equivalent, i.e., they generate the same topology with the metric given by d(x,y)=||x-y||. I'm confused about your second point, tho: if you have a chosen a norm, then the space must be normable.

 

[alexfloo]

You're entirely correct that they generate the same topology, but that doesn't make them equivalent. It makes the topologies induced from the norms equivalent. For most purposes in abstract linear algebra, the topology is all we are concerned with, but there are applications where it's the actual values of the norm as a function to the real numbers that are of interest, which is why I draw the distinction.

 

[Bacle2]

Right, good point. I had not thought of that.

 

[zooxanthellae]

Thanks for the many responses, I think I have a better picture of how this works now. A normed vector space is a vector space for which we have a defined norm, but a vector space without a norm is not necessarily a vector space for which a norm is undefinable?

 

[Citan Uzuki]

Right. I should add, since none of the other posters have done so already, that if you take any algebraic vector space without a topology, even an infinite-dimensional one, it is always possible to define a norm on that vector space. The proof is simple: every real vector space is isomorphic to a direct sum of (possibly infinitely or even uncountably many) copies of R, and the direct sum of any number of copies of R can be given the sup norm.