What is the significance of duality in vector spaces?

[wayneckm]

Hello all,

Recently I have read something about duality between vector spaces, however my intuition towards this is not clear. Wish someone can give me a hint.

Recall the definition of a dual pair is a 3-tuple $$(X,Y,\langle \cdot , \cdot \rangle)$$, so essentially duality between vector spaces is indeed the relationship of one vector space represents ALL (true?) continuous linear functional over the other, which is captured by $$\langle \cdot , \cdot \rangle$$.

So apparently $$\langle \cdot , \cdot \rangle$$ acts like an inner product while most text didn't define like this, so does that mean this can be of some form different from inner product? and can anyone name some example?

Furthermore, given this magical relationship, if we are just given one vector space $$X$$, is there any general rule that one can find $$Y$$ and $$\langle \cdot , \cdot \rangle$$? Because often most text would just give example of $$(X,Y,\langle \cdot , \cdot \rangle)$$ as a whole, rather than derving the remaining components with some given component in the 3-tuple. If not, this seems to force us to remember and study over some specific 3-tuples.

Thanks very much!

 

[Hurkyl]

The most obvious examples seem to be a self duality given by any inner product, and the duality between a vector space and its dual space.

I bet most natural "products" on infinite-dimensional spaces give rise to dualities too.


I imagine the whole reason to define the term is that there are specific interesting 3-tuples people like to study, and so they defined the abstract notion of a dual pair to simplify the exposition and to prove things about many examples at once.


I think all finite-dimensional examples are isomorphic to inner products. (or equivalently, isomorphic to the evaluation duality between V and V*)