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

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?)

• Yes looks good. Just note that you have $n$ variables so $x$ is a function of the form $f(x_1,x_2,...,x_n)=x_1$. – Yanko Feb 5 at 20:01