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 Jan 18 comment What does it mean for $f \in C^2(\mathbb{R}^2,\mathbb{R}^2)$? Can you give me a reference for the definition? Jan 18 asked What does it mean for $f \in C^2(\mathbb{R}^2,\mathbb{R}^2)$? Oct 11 awarded Enlightened Oct 10 awarded Nice Answer Jul 11 awarded Popular Question May 24 awarded Self-Learner Feb 26 awarded Yearling Jul 2 awarded Curious Jun 10 comment Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function Yes. Here I want a $C^k$, $k<\infty$ approximation, also I want the approximation in $L^1$. Given these milder restrictions (as opposed to $C^\infty$ and approximation in $L^\infty$) I wonder what is the best bound one can get for $\|f_\epsilon\|_{C^k}$. Jun 9 comment Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function Could you show me the calculation? Jun 5 asked Approximation of $C^\alpha(T^2)$ in $L^1$ by a $C^k$ function Jul 7 accepted The boundedness of $x^k e^{-|x|}$ Jul 4 comment The boundedness of $x^k e^{-|x|}$ Thanks @Ethan . Jul 4 asked The boundedness of $x^k e^{-|x|}$ Jul 3 asked approximate Fourier transform May 23 awarded Yearling May 9 awarded Caucus Apr 16 comment Bernoulli shift on $S^\mathbb{Z}$ First prove it for cylinders. Then prove it for finite unions of cylinders. I think then you can use an approximation argument: approximate (in symmetric difference) an arbitrary measurable set with finite union of cylinders. Apr 16 comment Nonconstant linear functional on a topological vector space is an open mapping You are assuming $f$ is continuous, which is not part of Rudin's statement. Apr 16 revised Nonconstant linear functional on a topological vector space is an open mapping added 21 characters in body