a strange iso revisited

hi there. after some (mild and sporadic) thinking about the question of finding a vector space V algebraically isomorphic to its linear space of endomorphisms, i’m getting to the point of saying that there’s no such V (this is a follow up of this post).

The reason is the following: if B is a basis of V, choose a vector w in it. Then consider the set P of nonempty parts of B, which do not contain w. Then P has cardinality strictly greater than B. Now consider the map F: P –> End(V) which associates to each N in P the (unique) endomorphism extending the function DN:B->V given by

D(v)=dN(v) .v + dW(v).w

where dN is the characteristic function of N in B, and dW the characteristic function of {w} in B. The mapping F is clearly injective. Now we notice that  every finite subset of Im(F) is linearly independent over k. Then Im(F) is linearly independent over k, it can be extended  to a basis of End(V).

This means that there’s a basis of End(V) with cardinality strictly greater than B, where B is a given basis of V. We conclude that ANY basis of End(V) has cardinality strictly greater than ANY basis of V.

Then, we have that V is never isomorphic to End(V), for ANY vector space V (over ANY field k).

Non-compact metric spaces

Last year, while on the bus with Laura after our Metric Spaces class - we started fooling around with the closure operator and cardinality. One of the questions that arose was this:

Given a metric space X, with cardinality card(X), it is possible to find for each cardinal b  a subspace B of X such that card( closure(B) ) = b ?

Also:

Let X be a metric space. Given a subspace B of X, is it possible to find a second subspace C of X such that closure (C) = B?

When B is a connected closed subspace*, that can be easily done using the interior of B; when B is an arbitrary subspace, i believe that it can’t be done even for connected spaces. I recall her trying to find a counterexample, but since all this happened last year, I don’t remember what she came up with… I believe it had something to do with the algebra of functions over X…. any ideas? The use/abuse of Zorn’s Lemma is welcome/allowed. As a consequence of that bus trip, we came up with an interesting Metric Space examination problem…

 

Prove: Every metric space of cardinality strictly greater than c is non-compact.

* Note: the connected requirement IS necessary, since for example in the complex plane, take B = B[0,1]   union {z=2}. Then interior(B)=B(0,1) and closure(interior(B))=B[0,1] which is a closed subspace properly contained in B.

- cheers.

manu.

A strange isomorphism

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