Some version of the theorem of Arzelà-Ascoli

Question

I have a question concerning the theorem of Arzelà-Ascoli. Let $(f_n)_n:[0,T]\to \mathbb{R}$ be a family of functions so that $(f_n)_n$ is uniformly bounded on $[0,T]$ and $(\frac{d}{dt}f_n)_n$ is uniformly bounded on $[0,T]$. In the paper I read $f_n$ is some integral operator for which they have shown $\frac{d}{dt}f_n\to \frac{d}{dt}f$ in $L^1((0,T))$. Then they conclude that by this $f_n\to f$ uniformly on $[0,T]$. Do we really need this last convergence? Due to https://en.wikipedia.org/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem it should be enough to have a uniform bound on the derivative which then gives $$|f_n(t)-f_n(\tilde{t})|\leq |\frac{d}{dt}f(\xi_{t,\tilde{t}})||t-\tilde{t}|\leq \|\frac{d}{dt}f\|_{L^{\infty}}|t-\tilde{t}|.$$ Does anybody have a reference for this situation?

Answer 1

Yes, we need. Arzela-Ascoli theorem gives the convergence of some subsequence. The reasoning is the following.

• Take any subsequence of our sequence.
• Use A-A theorem to prove that it has convergent sub-subsequence.
• Use your convergence in $L^1$ to prove that this sub-subsequence converges to $f$.
• You got that each subsequence has a sub-subsequence convergent to $f$. This implies that the whole sequence converges to $f$.

Answer 2

Just for convenience: The point seems to be that this is important to figure out the limit easy. If we know something about the convergence of the initial data one can just use $$\sup_{t\in [0,t]}|f_n(t)-f(t)|=\left|\int_0^t f_n'(s) ds+f_n(0)-\int_0^t f'(s)ds -f(0)\right|\leq \int_0^T |f_n'(s)-f'(s)|ds + |f_n(0)-f(0)|.$$