Definition 8.0.1.
If \(f_n:A\to V\) is a sequence of functions with a common domain, we say that \((f_n)_{n\in\mathbb{N}}\) converges pointwise to \(f:A\to V\) if for each \(x\in A\text{,}\) \(f_n(x)\to f(x)\) (as a sequence in \(V\)).
Chapter 8 Sequences of Functions
Now we’re going to talk about sequences of functions. But why would someone care about that?
Let’s take a function \(f\in \mathcal{C}^\infty([a,b])\text{.}\) Taylor’s Theorem tells us that, in some sense, we can approximate \(f\) by a polynomial \(f_n\) of degree \(n\text{,}\) with error controlled by \(f^{(n+1)}\text{.}\) It seems natural to ask: what happens as \(n\to\infty\text{?}\)
To answer this question, we’re going to have to discuss what it would mean for a sequence of functions to converge. We earlier discussed convergence in the context of a general normed space. Here, we’re going to work with vector spaces of functions like \(\mathcal{C}^k([a,b])\text{,}\) \(\mathcal{B}([a,b])\text{,}\) etc. The topological notions of open, closed, and compact will arise here, too.
Of course, in order for that topological approach to work, we will need to select norms on our various spaces of functions; or equivalently, to select a notion of convergence.
The most obvious idea of what it means for a sequence of functions \(f_n\) to converge to a limit \(f\) is:
It turns out, this notion isn’t so great.
