A very important theorem in linear algebra that is rarely taught is:

A vector space has the same dimension as its dual if and only if it is finite dimensional.

I have seen a total of one proof of this claim, in Jacobson's "Lectures in Abstract Algebra II: Linear Algebra". The proof is fairly difficult and requires some really messy arguments about cardinality using, if I remember correctly, infinite sequences to represent $\mathbb{N}\times\mathbb{N}$ matrices. Has anyone come up with a better argument in the 57 years since Jacobson's book was published, or is the noted proof still the only way to prove this fact?

A very important theorem in linear algebra that is rarely taught is:

A vector space has the same dimension as its dual if and only if it is finite dimensional.

I have seen a total of one proof of this claim, in Jacobson's "Lectures in Abstract Algebra II: Linear Algebra". The proof is fairly difficult and requires some really messy arguments about cardinality using, if I remember correctly, infinite sequences to represent $\mathbb{N}\times\mathbb{N}$ matrices. Has anyone come up with a better argument in the 57 years since Jacobson's book was published, or is the noted proof still the only way to prove this fact?

Edit: For reference, the proof is on pages 244-248 of Jacobson's

Lectures in Abstract Algebra: II. Linear Algebra.