Infinite dimensional vector space

In summary, an infinite dimensional vector space is a space with an infinite number of dimensions, represented by different vectors. Some examples include function spaces and sequence spaces. The main difference from a finite dimensional vector space is the number of dimensions, as an infinite dimensional space cannot be represented by a finite number of vectors. These spaces have applications in various fields and possess key properties such as a basis, dimension, subspaces, and topologies.
  • #1
Swati
16
0
Prove that \(R^{\infty}\) is infinite dimensional.
 
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  • #2
Swati said:
Prove that Rinfinity​ is infinite dimeensional.

Please be more specific about what you think \(R^{\infty }\) is?

(Try assuming otherwise and deriving a contradiction)

CB
 
  • #3
yes it is \(R^{\infty }\)
 
  • #4
Swati said:
Prove that Rinfinity​ is infinite dimeensional.

Suppose otherwise, that is that \( \mathbb{R}^{\infty}\) is finite dimensional with dimension \(N\)

Now consider \( \{e_1, e_2, ... , e_n, ... \}\) (where \(e_i\) is the element of \( \mathbb{R}^{\infty}\) with a zero in every position except for the \(i\)-th which is 1). Clearly \(\{ e_1, ..,e_N\}\) are linearly independent and therefore form a basis for \( \mathbb{R}^{\infty}\). But \(e_{N+1}\) cannot be written as a linear combination of the \(e_1, ... , e_N\) etc.

CB
 
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  • #5
Prove that \(F({\infty},-{\infty})\), \(C({\infty},-{\infty})\), \(C^{\infty}({\infty},-{\infty})\)
and \(C^m({\infty},-{\infty})\) are infinite dimensional.
 
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  • #6
Swati said:
Prove that \(F({\infty},-{\infty})\), \(C({\infty},-{\infty})\), \(C^{\infty}({\infty},-{\infty})\)
and \(C^m({\infty},-{\infty})\) are infinite dimensional.

Please provide context, what are these speces (try using words in addition to notation).

Presumably these are function spaces of some kind is so say so and which they are.

CB
 
  • #7
Prove that [FONT=MathJax_Math]F[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT], [FONT=MathJax_Math]C[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT], [FONT=MathJax_Math]C[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT]
and [FONT=MathJax_Math]C[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT] are infinite dimensional vector spaces.
(From Elementary Linear Algebra by Howard Anton)
 
  • #8
Swati said:
Prove that [FONT=MathJax_Math]F[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT], [FONT=MathJax_Math]C[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT], [FONT=MathJax_Math]C[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT]
and [FONT=MathJax_Math]C[/FONT][FONT=MathJax_Math]m[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]∞[/FONT][FONT=MathJax_Main])[/FONT] are infinite dimensional vector spaces.
(From Elementary Linear Algebra by Howard Anton)
You still have not explained what $F(-\infty, \infty)$ means (and as far as I know it is not a standard notation, so you should not expect it to be understood without an explanation).

For the spaces $C(-\infty, \infty)$ and $C^\infty(-\infty, \infty)$, let $f(x)$ be a nonzero $C^\infty$-function with support in the unit interval. For each integer $n$, define $f_n(x) = f(x-n)$. The functions $f_n$ form a linearly independent set and you can apply the Captain's argument in comment #4 above to show that these spaces are infinite-dimensional.

If the function $f$ can be chosen to be in the space $F(-\infty, \infty)$ (whatever that is), then the same approach will work to show that that space is also infinite-dimensional.
 

FAQ: Infinite dimensional vector space

What is an infinite dimensional vector space?

An infinite dimensional vector space is a mathematical concept that describes a space that has an infinite number of dimensions, where each dimension is represented by a different vector. This type of space is often used in abstract algebra and functional analysis.

What are some examples of infinite dimensional vector spaces?

Some examples of infinite dimensional vector spaces include function spaces, such as the space of all continuous functions on a closed interval, and sequence spaces, such as the space of all infinite sequences of real numbers.

How is an infinite dimensional vector space different from a finite dimensional vector space?

The main difference between an infinite dimensional vector space and a finite dimensional vector space is the number of dimensions. In a finite dimensional vector space, the number of dimensions is finite and can be represented by a finite number of vectors. In an infinite dimensional vector space, the number of dimensions is infinite and cannot be represented by a finite number of vectors.

What are some applications of infinite dimensional vector spaces?

Infinite dimensional vector spaces have many applications in mathematics, physics, and engineering. They are used to study and model complex systems, such as quantum mechanics and fluid dynamics. They are also used in data analysis and machine learning, where infinite dimensional vector spaces are used to represent high-dimensional data.

What are some key properties of infinite dimensional vector spaces?

Some key properties of infinite dimensional vector spaces include the existence of a basis, which is a set of linearly independent vectors that span the entire space, and the concept of dimension, which is the number of vectors in a basis. Additionally, infinite dimensional vector spaces can have subspaces that are also infinite dimensional, and they can have different topologies, which describe the ways in which vectors in the space can be added and multiplied.

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