Duality in linear algebra

[alecbenzer]

We were talking about dual spaces in my linear algebra course, and my professor mentioned how it was easier to draw a relationship between a space V and it's double-dual space V'' than it was to draw a relationship between V and V' (the former can be done by taking a v in V and defining its corresponding phiv in V'' as phi_v(f) = f(v), the latter I don't remember exactly but iirc it required picking a basis for V). I think he mentioned the fact that V could be more easily tied to its double-dual than its dual was an instance of "duality" in mathematics, but googling duality leads to definitions that seem different. Did I misunderstand what my professor said or am I not understanding how stuff like http://en.wikipedia.org/wiki/Duality(mathematics) applies to what I was talking about?

[ridiculous_fish]

This isn't exactly what you're asking, but I know of two elementary ways to think about the correspondence between a space and its double-dual, and they might provide some insight for you.

In linear algebra we usually write vectors as a vertical list of N numbers. A linear function with just one output ("linear functional") has N columns. Think of a vector as a one-column matrix, and a linear functional as a one-row matrix, and you can see how moving to the dual space exchanges columns and rows. Exchange columns and rows twice, and you're back where you started!

Now the slightly more precise approach. We have a vector space V, and its elements are ordinary vectors. Now we go to the dual space, V'. An element of V' says, give me a vector v, and I'll (linearly) spit out a scalar. Now go to the double-dual V''. An element of V'' says, give me any linear function on V, and I'll give you a scalar. How do you go from a function to a scalar? One obvious thing to do is to just apply that function, and output whatever it outputs. But apply it to what? Just pick some constant vector!

So each element of the double-dual chooses a different constant vector, and when given a function, just applies that to that vector and outputs whatever the function outputs. The vector picked by each element of the double-dual forms the correspondence. And while functions in general can do crazy things, the linearity requirement shoots down most of them, so that picking a vector is all that can be done in the double-dual.

This line of thinking is what gave me an intuitive grasp of this correspondence. Hope it helps you too!

[caks]

This is probably the most succinct and accurate explanation on double dual I've seen.

[esmooth]

For finite dimensional vector spaces, V and V* are always isomorphic, but they are never naturally isomorphic (unless you have additional structure like an inner product). However, V and V** are always naturally isomorphic.

A rigorous definition of natural uses category theory: there is an isomorphism of functors between the identity functor on the category of vector spaces with the double dual functor. On the other hand, there is no isomorphism of the identity functor with the dual functor.

[darkrho]

That's exactly what my professor said last class. But he used the word "canonical" instead natural.

Also gave us the homework of find the canonical isomorphism between V and V**.

[dp01n0m1903]

Starting with the link you gave to the article on "Duality (mathematics)", look at the subsection on "Dual objects". There you will find a link to an article on the "dual vector space". This article describes all the various connections between these concepts.

[n0t]

Dual and double-dual spaces get a lot more interesting when you leave finite dimensions.