0

I'm reading some class notes on harmonic analysis and the following lemma is used to prove that the Schwartz space is complete:

Assume that $(f_n)_{n=1}^\infty \subseteq C^1(\mathbb{R}^d)$ converges uniformly to a function $f$ and that each sequence of partial derivatives $(\partial_j f_n)_{n=1}^\infty$ converges uniformly to a function $g_j.$ Then $f \in C^1(\mathbb{R}^d)$ and $\partial_j f= g_j$ for every $j.$

However, the lemma itself is not proven. I'd like to see a proof or a reference for one. Thanks.

sebpar
  • 395
  • When $n=1,$ this is a standard result which you can find in Rudin PMA for example. This result leads to the $n>1$ result. (Note: You are using $n$ in two different ways.) – zhw. Dec 07 '20 at 18:26
  • See https://math.stackexchange.com/q/2117828/27978 for example. – copper.hat Dec 07 '20 at 18:28
  • I don't see how it follows from the $n=1$ case. – sebpar Dec 07 '20 at 18:38
  • For each $j$, fix the values of the other coordinates, while allowing $x_j$ to vary. For each fixed set of the other coordinates $(f_n) \subseteq C^1(\Bbb R)$, and the whole thing is just the $1D$ result. Once you've done this for every $j$, you've proven $f\in C^1(\Bbb R^d)$. – Paul Sinclair Dec 08 '20 at 05:15

0 Answers0