While walking my way back from school the other day, the following question struck me: is it possible for a vector space to be isomorphic to its ring of endomorphisms?

Clearly, one knows that regardless of the field you’re working with, the answer is negative for finite dimensional vector spaces since if V is n dimensional, then End(V) is n-squared dimensional. But… what happens for infinite dimensional vector spaces?

For a couple of days, I was convinced that NO ISOMORPHISM could be established. I started to mentally play with nilpotents and idempotents in End(V)  trying to reach a contradiction: an incompatibility between the ring structure of End(V) and the sole vector space structure of V.

On friday night I went for a long walk (following Connes’ advice) in the city: it was nearly 2 am when  left my apartment and set a random course over the streets of Buenos Aires. And then - zas! I recalled a basic fact: to vector spaces over the same field are isomorphic if and only if any pair of basis have the same cardinality… (I know, I know… I should have started here) and then said to myself: “Well… there MAY be an isomorphism between these two bastards…”

So the whole deal is to show that if B is a basis of V and C is a basis of End(v), then there exists a bijection between B and C.

I’ve been doing some mental calculations and getting there, and as a first guess/hint, I believe that the cardinality of the field should be less than or equal to te cardinality of B…

(Think about it, I don’t wanna spoil it)

 

Feel free to speak your mind and possible answers!

 

Cheers

Manuco