MHB Infinite dimensional vector space

[Swati]

Prove that \(R^{\infty}\) is infinite dimensional.

 

[CaptainBlack]

Prove that R^infinity​ is infinite dimeensional.

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

(Try assuming otherwise and deriving a contradiction)

CB

 

[Swati]

yes it is \(R^{\infty }\)

 

[CaptainBlack]

Prove that R^infinity​ 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

 

[Swati]

Prove that \(F({\infty},-{\infty})\), \(C({\infty},-{\infty})\), \(C^{\infty}({\infty},-{\infty})\)
and \(C^m({\infty},-{\infty})\) are infinite dimensional.

 

[CaptainBlack]

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

 

[Swati]

Prove that [FONT=MathJax_Math]F[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]), [FONT=MathJax_Math]C[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]), [FONT=MathJax_Math]C[FONT=MathJax_Main]∞[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main])
and [FONT=MathJax_Math]C[FONT=MathJax_Math]m[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]) are infinite dimensional vector spaces.
(From Elementary Linear Algebra by Howard Anton)

 

[Opalg]

Prove that [FONT=MathJax_Math]F[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]), [FONT=MathJax_Math]C[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]), [FONT=MathJax_Math]C[FONT=MathJax_Main]∞[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main])
and [FONT=MathJax_Math]C[FONT=MathJax_Math]m[FONT=MathJax_Main]([FONT=MathJax_Main]∞[FONT=MathJax_Main],[FONT=MathJax_Main]−[FONT=MathJax_Main]∞[FONT=MathJax_Main]) 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.