Theorems of the type $D\lim_n f_n = \lim_n Df_n$ for non-Archimedean fields

Question

I'm kind of interested in seeing if there is a base-field agnostic setting for calculus, but I don't have much experience with the non-Archimedian case. Let $X,Y$ be Banach spaces over some non-discrete locally compact field $k$, $U\subseteq X$ open. Consider a statement of the form:

Theorem (template)

Let $f_n: U\to Y$ be a sequence of differentiable functions that satisfies some property $A$, then there is a (at minimum pointwise) limit $f:U\to Y$ of $f_n$ with $f$ differentiable and $Df=\lim_n Df_n$.

If $k$ is the real or complex numbers an example of such a condition $A$ is:

Example (property $A$)

$Df_n$ converges locally uniformly to some function $F: U\to L(X,Y)$ and for every connected component $U_i$ of $U$ there is some $x_i\in U_i$ with $f_n(x_i)$ convergent.

The only proof I know uses the real line-integral, which is not available in the non-Archimedean case. I don't know if the statement itself remains true for other fields and haven't been able to find any similar kind of statement for this case.

Question

Is there a useful property for which the theorem is true for any locally compact field?

Answer 1

Not sure if this is what you're looking for, but section 69 of W. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis (Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511623844) is about "convergent sequences of differentiable functions". I'll state the main theorem below, after a bunch of notation.

Note that for the notion of differentiability here, which is defined via the plain old differential quotient (cf. Can we define a derivative on the $p$-adic numbers?, $p$-adic differentiation), we need to be able to divide (differences of) values of the codomain by (differences of) values of the domain. So the domain has to be contained in a field, not just some Banach space. (Of course e.g. any complete extension field of $\mathbb Q_p$ is also a Banach space over $\mathbb Q_p$ and hence would work here; also, some of this looks like it could easily generalize to the codomain being a more general Banach space over $K$; just the domain $X$ should really be in $K$ itself.)

Notation

Let $K$ be a complete non-archimedean non-trivially valued field (like $\mathbb Q_p, \mathbb C_p, k((T)), ...$).

Let $X \subset K$ be non-empty and without isolated points.

For $f:X \rightarrow K$, let

• $\lVert f \rVert_\infty := \sup_{x \in X} \lvert f(x) \rvert$;
• $\Phi_1 f(x,a) := \dfrac{f(x)-f(a)}{x-a}$ for $x, a \in X$ with $x \neq a$ ("difference quotient function");
• for any $a \in X$, let $\lVert f \rVert^a := \sup (\lVert f \rVert_\infty, \sup_{x \neq a} \lvert\Phi_1 f(x,a)\rvert)$.

Let $BD(X \rightarrow K) := \{f:X \rightarrow K: f \text{ is differentiable and } \lVert f \rVert_\infty < \infty\}$ be the space of bounded differentiable functions on $X$ with values in $K$.

Proposition

(69.3 in Schikhof.) Let $(f_n)_n$ be a sequence of bounded differentiable functions (i.e. all $f_n \in BD(X \rightarrow K)$), and assume there exists a function $f:X \rightarrow K$ such that $$(\ast) \qquad \text{for all } a \in X, \lim_{n \to \infty} \lVert f-f_n \rVert^a = 0.$$ Then $f$ itself is bounded and differentiable, and the sequence of derivatives $f'_n$ converges (pointwise) to $f'$.

Scholia

1. Schikhof also mentions that for each $a \in X$, $\lVert \cdot \rVert^a$ is a norm on $BD(X \rightarrow K)$; further (prop. 69.2), that if a sequence $(f_n)_n$ in $BD(X \rightarrow K)$ is Cauchy with respect to all these norms for all $a \in X$, then there exists a function $f \in BD(X \rightarrow K)$ such that $(\ast)$ holds, i.e. in a way the space of bounded differentiable functions is complete with respect to this notion of Cauchy and convergence.

2. The criterion $(\ast)$, which is my candidate in this setting for what you call property A, is obviously stronger than uniform convergence $\lVert f-f_n\rVert_\infty \to 0$. As often in the ultrametric setting, one has to also "bound" (and, for that matter, "bound uniformly") the "difference quotient function" $\Phi_1 f$.

3. Relatedly, Schikhof shows (prop. 69.5) that the following are equivalent for a sequence $(f_n)_n$ of bounded differentiable functions:

(i) For all $a \in X$, $(f_n)_n$ is Cauchy with respect to $\lVert \cdot \rVert^a$.

(ii) $\lim_{n \to \infty} f$ exists uniformly and pointwise, and for each $a \in X$ and $\varepsilon > 0$, there exists $\delta > 0$ such that for all $n \in \mathbb N$, $0 < \lvert x-a \rvert < \delta$ implies $\lvert \Phi_1 f_n(x,a) - f'_n(a) \rvert < \varepsilon$.

He calls the property in $(ii)$ "equidifferentiable"; as you see, we somehow need that at each point, all the derivatives of our function sequence are uniformly approached by the respective difference quotients.

4. Note that if for $X$ you take something compact like $\mathbb Z_p$, you get the boundedness, and a lot of things to check above, for free.

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Added later: I overlooked that Schikhof states (actually, leaves as exercise to the reader) a preliminary version of the above, much earlier in the book. Namely, a function $f$ like above is called a $C^1$-function (which implies but is stronger than having a continuous derivative, see above links) if for every $a\in X$ the limit $$\lim_{(x,y) \rightarrow (a,a), x \neq y} \Phi_1 f (x,y)$$ exists. Then :

(Exercise 27.C. in Schikhof.) Let $(f_n)_n$ be a sequence of $C^1$-functions which converges (pointwise) to a function $f$, and assume that

$$(\ast \ast) \qquad \lim_{n \to \infty} \Phi_1 f_n \; \text{ exists uniformly on } \; X \times X \setminus \{(x,x):x \in X\}$$

Then $f$ itself is a $C^1$-function, and the sequence of derivatives $f'_n$ converges (uniformly) to $f'$.

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Finally, note that there are classical counterexamples to attempts of getting statements like above with conditions weaker than $(\ast)$ and $(\ast \ast)$. Schikhof asks the reader to construct one as follows in Exercise 26.F: If we write every $x \in \mathbb Z_p$ ($p$-adic integers) as a sum $\sum_{k \ge 0} a_k(x)p^k$ with uniquely determined $a_k(x) \in \{0,...,p-1\}$, and call $f_n: \mathbb Z_p \rightarrow \mathbb N (\subset\mathbb Q_p \subset \mathbb C_p)$ the function which cuts off the series after the $n$-th term,

$$ f_n: x \mapsto \sum_{k=0}^n a_k(x)p^k ,$$

then each $f_n$ is locally constant, in particular the sequence of derivatives $f'_n$ is just constant $=0$ and converges uniformly to $0$; however, the sequence $f_n$ actually converges uniformly to the identity function $f: x \mapsto x$, which of course has derivative $f' = 1$. Note that all functions involved here are actually $C^1$, but it's really the "difference quotient" functions $\Phi_1 f_n$ which make the failure of the conditions $(\ast)$ or $(\ast \ast)$ above in this example crucial.