I'm studying by myself PDEs without having done Functional Analysis and I'm trying do the following exercise of the book "Partial Differential Equations I" by Michael E. Taylor on Appendix $A$ - Section $6$ (page $592$):
- Prove the following result, also known as part of Ascoli's theorem. If $X$ is a compact metric space, $B_j$ are Banach spaces, and $K: B_1 \longrightarrow B_2$ is a compact operator, then $\kappa f(x) = K(f(x))$ defines a compact map $\kappa: \mathcal{C}^{\alpha}(X,B_1) \longrightarrow C(X,B_2)$, for any $\alpha > 0$.
$\mathcal{C}^{\alpha}$ denotes a space with $\alpha$-Holder continuity (this notation was introduced on the final of the page $317$).
I would like to know if my attempt it's correct until where I wrote and, if it is correct, how I can ensure the limit below lives in $C(X,B_2)$.
$\textbf{My attempt:}$
Firstly, we observe that $\kappa$ is well-defined, otherwise, $K$ wouldn't well defined, which would be an absurd since $K$ is a map.
Now, given a bounded sequence $(f_n)$ in $\mathcal{C}^{\alpha}(X,B_1)$ and fixing $x \in X$, we have that $(f_n(x))$ is a bounded sequence in $B_1$. By compactness of the map $K$, there is a subsequence $(f_{n_l}(x))$ in $B_1$ such that $K(f_{n_l}(x)) \rightarrow K(f(x)) \in B_2$, i.e., given $\varepsilon > 0$, there is $N(x) \in \mathbb{N}$ such that
$$l > N(x) \Longrightarrow ||K(f_{n_l}(x)) - K(f(x))||_{B_2} < \frac{\varepsilon }{3}\ (*)$$
The subsequence $((K \circ f_{n_l}))$ is in $C(X,B_2)$ since $K$ and $f_{n_l}$ are continuous functions. Since the limit is unique, $f$ is well-defined.
By continuity of $K \circ f_{n_l}$ in every $x \in X$, there is $\delta_x > 0$ such that
$$||y - x||_X < \delta_x \Longrightarrow ||K (f_{n_l})(y) - K (f_{n_l})(x)||_{B_2} < \frac{\varepsilon}{3} \ (**)$$
W.l.o.g., we assume $\frac{1}{N(x)} < \delta_x$. By compactness of $X$, there is a finite subcover of $X$ by open balls $B(x_i,\frac{1}{N(x_i)}) \subset X$, $i = 1, \cdots, j$. Thus, arguing as before, we find $(K \circ f_{n_l}(x))$ such that $(*)$ holds for $N := \max_\limits{ i \in \{ 1,\cdots,j \} } N(x_i)$. Denoting by $\delta := \frac{1}{N}$ and using $(*)$ and $(**)$, we have that
$||K(f(y)) - K(f(x))||_{B_2} \leq ||K(f(y)) - K(f_{n_l}(y))||_{B_2} + ||K(f_{n_l}(y)) - K(f_{n_l}(x))||_{B_2} + ||K(f_{n_l}(x)) - K(f(x))||_{B_2}$
$\begin{eqnarray*} ||K(f(y)) - K(f(x))||_{B_2} &\leq& ||K(f(y)) - K(f_{n_l}(y))||_{B_2} + ||K(f_{n_l}(y)) - K(f_{n_l}(x))||_{B_2} + ||K(f_{n_l}(x)) - K(f(x))||_{B_2}\\ &\leq& \frac{\varepsilon}{3} + \frac{\varepsilon}{3} + \frac{\varepsilon}{3} = \varepsilon, \end{eqnarray*}$
whenever $l > N$ and $||y - x||_X < \delta$, hence $(K \circ f) \in C(X,B_2)$. $\square$
I'm stuck here, because I can't prove that $(K \circ f) \in C(X,B_2)$. I know that $(K \circ f_{n_l}) \in C(X,B_2)$ for each $l \in \mathbb{N}^*$, $(K \circ f_{n_l})(x) \rightarrow (K \circ f)(x)$, $(K \circ f_{n_l})$ is bounded I know the uniform limit of continuous functions is continuous, but I can't see how I can prove the pointwise convergence is, indeed, uniform. The only thing that I thought about this is use Ascoli's theorem because I know that $(K \circ f_{n_l})$ is bounded, but I can't see how the family $\{ (K \circ f_{n_l}) \in C(X,B_2) \ ; \ l \in \mathbb{N}^* \}$ is equicontinuous.
I thought about the MaoWao's comment and I think I'm close to solving the question, but I don't sure if $(*)$ by the $N$ that I chose. If this is not the right $N$ that I need to take, someone can help me with a hint about how I can choose the $N$ without depending on $x$?
Thanks in advance!
$\textbf{EDIT:}$
The version of Ascoli's theorem that I'm using:
$\textbf{Ascoli-Arzela's theorem:}$ let be $E$ a set of continuous maps $f: K \longrightarrow N$, where $K$ is compact. $E \subset C(K,N)$ is relatively compact if, and only if, the following holds:
1) $E$ is equicontinuous;
2) For each $x \in K$, $E(x) = \{ f(x) \ ; \ f \in E \}$ is relatively compact in $N$.
This will follow pretty directly from the version of Ascoli-Arzela that you already know.
Let $D$ be the unit ball of $C^\alpha(X, B_1)$. Your goal is to show that $E = \kappa D$ is relatively compact, i.e. that the set $E = \{\kappa f : f \in D\}$ is relatively compact in $C(X, B_2)$ (here $N = B_2$). You want to verify the hypotheses of Ascoli-Arzela.
For (1), use the Hölder continuity. You should be able to show that for any $f \in D$ and any $x,y \in X$, we have $\|(\kappa f)(x) - (\kappa f)(y)\|_{B_2} = \|K(f(x)) - K(f(y))\|_{B_2} \le \|K\| d(x,y)^\alpha$, where $\|K\|$ is the operator norm of $K$ and $d$ is the metric on $X$. Equicontinuity should then follow easily.
For (2), fix $x \in X$ and note that for any $f \in D$, we have $\|f(x)\|_{B_1} \le 1$. Now use the fact that $K$ is a compact operator to conclude that $\{(\kappa f)(x) : f \in D\}$ is relatively compact in $B_2$.