Infinite vector space

1. Jul 12, 2006

Awatarn

How could I proof that $$F^\int$$ is infinite vector space?

2. Jul 12, 2006

matt grime

Just show there is an infinite set of linearly independent elements.

We can't say any more than that unless you define what F^\int actually is.

3. Jul 12, 2006

HallsofIvy

Staff Emeritus
You mean "infinite dimensional" vector space. Every vector space is infinite! But I agree with matt grime: what is $F^\int$?

4. Jul 12, 2006

mathwonk

well every positive dimensional vector space over an infinite field is infinite, but {0} is a finite subspace of every vector space, and no finite dimensional vector space over Z/2 is infinite.

5. Jul 12, 2006

HallsofIvy

Staff Emeritus
Okay, Okay! I knew when I wrote that that someone would clobber me!

6. Jul 12, 2006

mathwonk

did not intend to clobber you, merely correct the statement. most of my statements require several iterates to get right.

7. Jul 23, 2006

Awatarn

ohhhh! I'm sorry. $$F^\int$$ should be $$F^\infty$$

8. Jul 23, 2006

matt grime

It would still be a good idea to state what that means. We can guess (correctly, I imagine, that F is some field, and you mean the infinite direct sum of that field with itself countably many times), but we shouldn't have to.

9. Jul 24, 2006

Awatarn

I found this problem in my textbook "Linear Algebra Done Rigth." It states that "Proove that $$F^\infty$$ is infinite dimensional vector space."
$$F$$ is some field, $$\matcal{R}$$ or $$\matcal{C}$$.

I understand that this field with itself is countably many times, but how can I proove in beuatiful mathematical language.
In my opinion, this problem is similar the followed problem. how you could show that every cormorant bird are black. then you try to catch all of them as many as you can, but you could not found the other color of cormorants. therefore you conclude that every cormorants is black. this method is not an easy and beautiful way.
How should I do?

10. Jul 25, 2006

matt grime

It has already been explained what you need to do. Post 2.

11. Jul 31, 2006

mathwonk

here is a possible definition of F^infinity: say it consists of all real valued functions on the unit interval. prove it is infinite dimensional as a real vector space. (please do not quibble that this is an unlikely definition.) it still suffices for the main idea here.