Difference between points and vectors in $\mathbb{R}^n$ in relation to vector-valued functions

Question

I'm very new to vector calculus, and based on what I read I may be dealing with a trivial distinction, but I want to be completely sure of this.

I know that $\mathbb{R}^n$ contains points, sometimes denoted $\mathbf{x}$, and vectors, denoted $\mathbf{\vec{v}}$. It seems that when we consider a function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$, we are mapping points in $n$-dimensional space to points in $m$-dimensional space. Similarly, when we consider a limit of this function, it seems that we are interested in the behavior of the function values as points in our domain approach some other point. However, the common terminology seems to be that $\mathbf{f}$ is a "vector-valued function": this would seem to suggest that we are outputting vectors. I have also seen it stated that we are considering the behavior of vectors in the domain of the function when taking limits.

I've read that there is often no distinction made between points and vectors in this respect. I've also seen some conflicting information on whether points are simply "vectors through the origin," but I am having difficulty understanding what is really at work with these functions.

Thanks in advance for any helpful insights.

Answer 1

In general, if $x\in\Bbb R^n$ (or $x\in S$), then $x$ is just that: an element of the set $\Bbb R^n$ (or $S$). Depending on context, we may call $x$ a point, a vector, an $n$-tuple, or perhaps a vehicle (e.g., if. $S=\{\text{car},\text{bicycle},\text{truck}\}$). The term "point" is used when we focus on geometry or topology properties of $\Bbb R^n$, the term "vector" when we focus on $\Bbb R^n$ being a vector space, i.e., when we need to know how to add elements or multiply them with a constant. Topology comes into play with $\Bbb R^n$, whenever convergence does.

In the end, $\Bbb R^n$is a "topological vector space", i.e., not only a topological space and a vector space at the same time, but it also plays both these roles in a compatible way (i.e., the vector space operations of adding two vectors or scaling a vector are continuous maps). At any rate, "point" or "vector" are just synonyms for "element" with a bit of contextual connotation added. When we refer to a map to $\Bbb R^m$ as "vector-valued", we want to emphasize that the co-domain is not just $\Bbb R$ and that therefore that we may need to use different tools.

Answer 2

$\Bbb R^n$ is a vector space (just check the axioms of a vector space), and its elements are therefore called vectors.

However, it can also be seen as an affine space and its elements are then called points.

An affine space $E$ is more or less like a vector space where you "forget the origin": you can get vectors starting from any point $A$, and given two points $A$ and $B$, you can get a vector $\vec{AB}$.

Actually, an affine space has an associated vector space, and the vector space associated to the affine space $\Bbb R^n$ is... $\Bbb R^n$, and that's why this may look confusing. The elements are thus called points or vectors depending on the structure that you want to emphasize. If you don't rely to much on the structure, you can use both terms.