What does it mean for a sequence of curves to converge to another curve?

Question

I can easily define what it means for a sequence of points of $\mathbb{R}^n$ to converge to another point in $\mathbb{R}^n$. But what does it mean for a sequence of curves in $\mathbb{R}^n$ to converge to another curve in $\mathbb{R}^n$? What is the formal definition of it? I am asking because I am looking for a formalization of the statement "As $n$ approaches $\infty$, an $n$-sided regular polygon approaches a circle", and similar statements.

Answer 1

So, there are several ways of doing this, so here is a couple:

Let $I$ be some predefined interval. First, you can say a sequence of curves $\gamma_n: I \rightarrow R^n$ converges to $\gamma: I \rightarrow R^n$ if for all $x \in I$, $\gamma_n(x)$ goes to $\gamma(x)$. This is commonly referred to as pointwise convergence.

Another, more restrictive form of convergence (that has some nicer properties) is called uniform convergence: for all $\epsilon > 0$, there exists some $N \in \mathbb{N}$, such that for all $n > N$, the supremum of $|\gamma_n(x) - \gamma(x)|$ is less than $\epsilon$. Intuitively, the difference between the two is that uniform convergence doesn't just mean every point converges, but they converge at a similar pace. If $I$ is compact, the two notions are the same.

Finally, there is a third way to think about this: This works for any collection of compact subsets of $\mathbb{R}^n$, and does not rely on parameterization. Basically, if $A,B \subset \mathbb{R}^n$ are compact, we can define their distance as, and this is going to look complicated, $\max(\sup_{x \in A}(\inf_{y \in B}|x-y|), \sup_{x \in B}(\inf_{y \in A}|x-y|))$. You can prove that this defines a metric on the set of all compact subsets of $\mathbb{R}^n$, so if you just restrict your attention to compact sets that can be written as the image of a smooth (or piecewise smooth, or any other class you want) embedding of $[0,1]$ into $\mathbb{R}^n$, then this is a formal way to define distance between them without having to worry about parameterization.