Proving F^\int as an Infinite Vector Space

[Awatarn]

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

 

[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.

 

[HallsofIvy]

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

 

[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.

 

[HallsofIvy]

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

 

[mathwonk]

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

 

[Awatarn]

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

 

[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.

 

[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?

 

[matt grime]

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

 

[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.