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Please be more specific about what you think \(R^{\infty }\) is?Prove that Rinfinity is infinite dimeensional.
Suppose otherwise, that is that \( \mathbb{R}^{\infty}\) is finite dimensional with dimension \(N\)Prove that Rinfinity is infinite dimeensional.
Please provide context, what are these speces (try using words in addition to notation).Prove that \(F({\infty},-{\infty})\), \(C({\infty},-{\infty})\), \(C^{\infty}({\infty},-{\infty})\)
and \(C^m({\infty},-{\infty})\) are infinite dimensional.
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).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)