# price space as a dual of commodity space

I often see papers in economic theory mentioning a duality between the price space and the commodity space. I think they refer to duality of vector spaces. I.e, the commodity space is a vector space, a price system is a linear functional over the commodity space (assigning a value to each bundle), and so the space of all possible price systems is the dual of the commodity space.

Is there a standard reference that explains this duality in more detail?

• Not sure about the standard reference. It seems to me a standard functional analysis textbook will provide you with a lot of details. But sometimes, a book on mathematical economics, say Michael Carter's foundation, will suffice. Also, financial economics also rely on this structure a lot. Commented Apr 13, 2015 at 11:27

If you want to see a nice usage of duality in economics, in Mas-Colell, Whinston, and Green there's a nice section on duality in the second or third chapter when talking about Hicksian demand. The treatment is pretty straightforward if the consumption space is real and has a finite dimension. I think understanding that basic usage of duality is illustrative of its general utility.

If the idea is to understand what duality is and how it operates, I think Conway is a great book.

(Just undergrad here, hope I'm not mixing up topics).

• What you talk about is relevant, of course, and could serve as a intuitive starting point. But as a finite dimensional topological vector space you mentioned, the consumption space is reflexive. Weak topology and strong topology coincide. Regarding OP's previous questions, it seems to me, this reference is not something what he really need. Commented Apr 14, 2015 at 12:39
• Oh I seem to have understood the question differently. I thought he was looking for a clear usage of duality in economics. I think Conways analysis book is really good if the idea is to understand duality.
– BVJ
Commented Apr 14, 2015 at 12:48
• Yes. It's also possible that I understand OP in the wrong way. Maybe you want to add that point in your answer to make it more comprehensive. Commented Apr 14, 2015 at 12:53

For a nice, and gentle, functional analytic treatment on this, see the chapter on welfare theorems in Stokey and Lucas. (Although what they call "inner product" is definitely not standard functional analytic language. That term is reserved strictly for Hilbert spaces in functional analysis.)

Prices are elements of the dual as you and previous answer stated. Where one needs to be more careful in the general setting is specifying a topology. You want the value of bundles to be continuous---hence prices are continuous (i.e. bounded, if the commodity space is Banach) with respect to the topology chosen on the commodity space.