Double duals

Double duals

[Tom Leinster]

Dear all,

I'm looking for occurrences in the literature of two theorems about double
duals of vector spaces.  In both cases, I can prove the theorem but want
to know who to attribute it to.  I'd be grateful if anyone could tell me.

Let Vect be the category of all vector spaces over some field.
Dualization defines a contravariant functor from Vect to Vect.  It's
self-adjoint on the right, making double dualization into a monad on Vect.

Here are the theorems.

1. The double dualization monad is the codensity monad of the inclusion
FDVect ---> Vect (where FDVect is the category of finite-dimensional
vector spaces).

2. The category of algebras for the double dualization monad is Vect^op;
that is, the single dualization functor Vect^op --> Vect is monadic.  I
learned this from a MathOverflow answer of Todd Trimble:

    http://mathoverflow.net/questions/104777

I understand that Fred Linton proved something similar for Banach spaces,
but I'm principally interested in the result for unadorned vector spaces.

Thanks very much,
Tom
































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