I'm reading Gong Sheng's Concise Complex Analysis, where it introduced a Weierstrass Theorem
Theorem 3.1 (Weierstrass Theorem) Suppose $\{f_n(z)\}$ is a sequence
of functions where each $f_n(z)$ is defined and holomorphic in a
region $U\subseteq \mathbb C$. Assume that $\sum_{n=1}^\infty f_n(z)$
converges uniformly to $f(z)$ on every compact subset of $U$.
Then $f(z)$ is holomorphic on $U$ and for every $k\in \mathbb N$,
$\sum_{n=1}^\infty f_n^{(k)}(z)$ converges uniformly to $f^{(k)}(z)$
on every compact subset of $U$.
Then it mentions:
This is a profound result. The reader can compare it with the theorem
of the derivative of function series in calculus.
So what is the corresponding real analysis version of this complex analysis Weierstrass Theorem, and what is the difference? — I suppose the difference would show some distinct properties in complex analysis.
Best Answer
We find in section 10.2.3 Uniformly Convergent Series of Concise Calculus by Sheng Gong the following theorem:
Conclusion: Here we see how strong the concept of a holomorphic function is. In the Weierstrass theorem we consider functions $f_n$ being holomorphic on a compactum. This implies also the existence of derivatives of the functions $f_n$ of arbitrary order.
Armed with that it is sufficient to require uniform convergence of the holomorphic functions $f_n$ on a compactum and it follows not only that $f(z)=\sum_{n=1}^\infty f_n(z)$ is holomorphic on this compactum but also the uniform convergence of all $k$-th derivatives $f^{(k)}$ of $f$ on it.
On the other hand, in the real case we have to require that all the functions $u_n(x)$ are $C^1$, i.e. differentiable and the derivatives are continuous. Contrary to the complex case we have to require that the derivatives are uniformly convergent on $[a,b]$ and we cannot conclude anything for higher derivatives of the functions $u_n(x)$, since we even don't know if they exist.