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Prove that spaces $C^k(\mathbb{R}^n)$ and $C^\infty(\mathbb{R}^n)$ are infinite dimensional.

So in order to prove that they're infinite dimensional spaces, I need to form a linearly independent sequence which is infinite. Would I simply pick polynomials? (i.e. $1,x,x^2,\dots$ as my basis?)

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  • $\begingroup$ Yes looks good. Just note that you have $n$ variables so $x$ is a function of the form $f(x_1,x_2,...,x_n)=x_1$. $\endgroup$ – Yanko Feb 5 at 20:01
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Yes and no. Yes, because they form an infinite linearly independent family, and that is enough to prove what you want to prove. And no because they do not form a basis.

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