Proving F^\int as an Infinite Vector Space

In summary, the conversation discusses how to prove that F^\infty, an infinite dimensional vector space, is infinite. It is suggested to show that there is an infinite set of linearly independent elements. However, in order to do this, the definition of F^\infty must be clarified as it is not explicitly stated. It is also mentioned that this problem is similar to proving that all cormorant birds are black by observing a large sample. One possible definition of F^\infty is given as all real-valued functions on the unit interval. Ultimately, the main idea is to prove that F^\infty is infinite dimensional as a real vector space.
  • #1
Awatarn
25
0
How could I proof that [tex]F^\int[/tex] is infinite vector space?
 
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  • #2
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
You mean "infinite dimensional" vector space. Every vector space is infinite! But I agree with matt grime: what is [itex]F^\int[/itex]?
 
  • #4
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
Okay, Okay! I knew when I wrote that that someone would clobber me!
 
  • #6
did not intend to clobber you, merely correct the statement. most of my statements require several iterates to get right.
 
  • #7
ohhhh! I'm sorry. [tex]F^\int[/tex] should be [tex]F^\infty[/tex]
 
  • #8
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
I found this problem in my textbook "Linear Algebra Done Rigth." It states that "Proove that [tex]F^\infty[/tex] is infinite dimensional vector space."
[tex]F[/tex] is some field, [tex]\matcal{R}[/tex] or [tex]\matcal{C}[/tex].

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
It has already been explained what you need to do. Post 2.
 
  • #11
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.
 

1. What is an infinite vector space?

An infinite vector space is a mathematical concept that describes a collection of vectors that can be added and scaled infinitely. In other words, there is no limit to the number of vectors that can exist in this space.

2. How is an infinite vector space different from a finite vector space?

A finite vector space has a limited number of vectors, while an infinite vector space has an infinite number of vectors. Additionally, a finite vector space has a finite dimension, while an infinite vector space has an infinite dimension.

3. What are some examples of infinite vector spaces?

Some examples of infinite vector spaces include the space of all polynomials, the space of all continuous functions, and the space of all sequences.

4. Can an infinite vector space have a finite basis?

No, an infinite vector space cannot have a finite basis. This is because a basis is a set of linearly independent vectors that can be used to span the entire vector space. In an infinite vector space, there is no finite set of vectors that can span the entire space.

5. How is an infinite vector space used in real-world applications?

Infinite vector spaces are used in various fields of science, including physics, engineering, and computer science. They are used to model continuous phenomena and to solve complex problems that involve infinite dimensions, such as optimization and data analysis.

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