# Infinite dimensional vector spaces without basis?

1. Oct 20, 2012

### Aziza

According to my professor, there exist infinite dimensional vector spaces without a basis, and he asked us to find one. But isn't this impossible? The definition of a dimension is the number of elements in the basis of the vector space. So if the space is infinite-dimensional, then the basis of that space has an infinite amount of elements..

the only vector space I can think of without a basis is the zero vector...but this is not infinite dimensional..

2. Oct 20, 2012

### pwsnafu

Err, the statement "every vector space has a basis" is equivalent to the axiom of choice.

3. Oct 21, 2012

### Stephen Tashi

It's not impossible if your course materials say that every element in a vector space with a basis must be representable as a finite linear combination of the basis vectors or define the concept of "linear combination" as a finite sum. A basis for a vector space is usually defined that way. You'll have to read the fine print in how your materials define a vector space. It may be that "infinite dimensional" merely means "not finite dimensional" instead of implying that there is an infinite basis.

There are various modified definitions of a basis that are applicable to infinite dimensional vector space. See the "related notions" section of the Wikipedia article http://en.wikipedia.org/wiki/Basis_(linear_algebra).

Last edited: Oct 21, 2012
4. Oct 21, 2012

### Fredrik

Staff Emeritus
As the two guys before me said, with the standard definitions, every vector space has a basis (assuming that we're working within the branch of mathematics defined by ZFC set theory).

Someone else asked about how to prove that every vector space has a basis a couple of weeks ago. This is what I said to him:

5. Oct 22, 2012

### mathwonk

we are saying the statement you ascribe to your professor is false in the presence if the usual axiom of choice, so you should verify what he said with him.

and it is easy to define an infinite dimensional space as one with no finite basis, without allowing a basis for the big one, and also the basis of {0} is the (finite) empty set.

6. Oct 23, 2012

### Vargo

Perhaps he meant you to find a space without a countable basis.