Is there a convention to interpret equalities of functions as series?

Question

I am wondering about whether there is a default or standard interpretation of statements such as $$\sum_{n=1}^\infty f_n(x) = f(x)$$ or equivalently

$$\sum_{n=1}^\infty f_n = f$$

In some cases these statements can mean 'uniformly convergent to $f$' or just 'pointwise convergent to $f$'. But sometimes I come across these equalities without the uniform or pointwise qualification, and thus in these situations I don't know whether as a default to interpret them as meaning pointwise or uniform convergence.

For example, when I first learnt about power series, we had not yet met the notions of uniform convergence (or pointwise). We simply defined $f(x) = \sum_{n=1}^\infty a_nx^n$. In hindsight, this equality really is equivalent to asserting the pointwise convergence of the series to $f$ over the radius of convergence. (Although it also turns out to be uniformly convergent within the radius)

Another example comes from the second answer in this question: When can a sum and integral be interchanged?, from the user Jonas Teuwen. In particular, he states that $f = \sum_n f_n$ in his answer. How should these equalities be interpreted? Is there a default, e.g. just assume it means pointwise, or is it entirely context dependent?

[Note: my current understanding is that when we deal with infinite series of functions, writing it as an equality is really a shorthand for some first order logic statement. I.e. it is completely analogous to the fact that stating $\lim_{n \rightarrow \infty} a_n = l$ in the case of real sequences really means $\forall \epsilon >0 \exists N \forall n>N (|a_n - l|< \epsilon)$. In this way I think of the equality symbol as just shorthand for a more verbose expression when it comes to series of functions, rather than meaning equality of mathematical objects so to say. In this sense, I don't know how to interpret the statements abut equality of series without any context.]

Answer 1

By default, it means that the convergence is pointwise. If it's meant to be uniform convergence, then it's either mentioned, or you are working in a function space with a topology where convergence means uniform convergence.

Like the normed space $(C^0(K),\Vert\cdot\Vert_\infty)$ of continuous functions $K\longrightarrow\mathbb R$ on a compact set $K$ with the norm $\Vert f\Vert_\infty:=\sup_{x\in K}\vert f(x)\vert$. Convergence in this normed space is equivalent to uniform convergence.

Or you could consider the topological space $(\mathcal H(\mathbb C),\tau)$ of holomorphic functions on $\mathbb C$ with the topology of compact convergence $\tau$. There, $\sum_n f_n=f$ would usually be interpreted to mean compact convergence.

So essentially, if it's explicitly mentioned that you're working in a function space whose topology implies a specific kind of convergence, then that kind of convergence is probably implied. Otherwise, the default is pointwise.