In this post I’m going to describe a (probably ridiculous) idea I came up with, which seems to only be interesting for infinite-dimensional vector spaces.
Recall that given a vector space over a field , the dual space is defined as follows:
It therefore makes sense to talk about the dual of any vector space. Even has a dual, which we denote . It’s defined exactly the way is defined, except of course we replace all the occurrences of with . So why do we call this set a dual space? Well, we can turn it into a vector space over by making some no-brainer definitions:
- For all , we define the function by for all .
- For all and , we define the function by for all .
I’ll leave it to you to check the axioms 🙂 but it does indeed work. Anyways, my idea takes an interest in the relationship between and its double-dual , because there is a natural way we can send the elements of into . How, you ask? Well, we proceed as follows. For each vector , define the evaluator of by for all . That is, the evaluator of takes in an element of (a linear functional on ) and spits out the result of evaluating that functional at . Pretty straightforward, I think. Verify yourself that .
Now, we want to find a map from into . What better way to do this than to send each vector to its evaluator ? Therefore, define by putting for all . It’s completely natural — depends on no choice of basis at all! This is wonderful!
It turns out that this map is linear, and it’s always an injection. Here’s why my idea only involves infinite-dimensional vector spaces: for finite-dimensional vector spaces, the map turns out to be a vector space isomorphism. In other words, and turn out be structurally identical. This is boring.
The basic idea I’ll describe in words: I want to examine the vector spaces , , and so on. Of course, is not a subspace of , heck, it’s not even a subset — but we have this map which we can use to “embed” vector spaces into their double-duals. So really, I mean , , and so on.
In a more “proper” (aka terse) way of speaking, define and recursively put for . Now for each , define the map by putting for all . Put . What can be said about the sequence ?
Note the reason it’s “boring” in the finite-dimensional case is because all the spaces end up being trivial… lol. But in the infinite-dimensional case it seems like there are many questions we can ask. It’s odd because using this construction, we come up with a quotient space that is naturally associated with …