How to denote an $n$-dimensional point

Question

An $n$-dimensional point:

$\textbf{x} = (x_1, x_2, \ldots , x_n)$, for some $n \in \mathbb{Z}, n > 1$

In the book these are denoted by bold letters as above.

It's hard for me to write bold letters on paper. What are some quick and recognizable notational conventions for $n$-dimensional points which translate to handwritten work.

Any tips appreciated.

Answer 1

It depends on the context. In a geometry context, there is an equivalence between points in space and vectors, so the notation $\vec{x}$ can be used.

In a matrix-related context, generally capitals are used, like $X$.

Answer 2

Personally, I don't like using $\bar{x}$ as it reminds me of the mean value of a sample $x=(x_1,x_2,\dots,x_n)$.

On the other hand, the arrow $\vec{x}$ notation is too vector-based in my opinion, but if that's what you need it's great.

I prefer using $\mathbb{x}$ to represent an $n$-tuple in my writing, so $$\mathbb{x}=(x_1,x_2,\dots,x_n)\in R^n.$$

Doubled lines look great on paper, and I prefer it to $\vec{x}$ as a function argument. For instance, I would write $$f(\mathbb{x})=\frac{1}{n}\sum_{i=1}^n x_i,$$ and denote such a function with $\bar{\mathbb{x}}$.

I found this notation in tensor calculus, and it was great since the capital letters (such as $X$) were reserved for random variables.

Answer 3

Minor, pedantic nitpick: a point is zero dimensional. The objects you are describing are points in an $n$-dimensional space. The points themselves are not $n$-dimensional.

With respect to notation, it sounds like you are writing notes or completing homework assignments. In that kind of context the notation can be a lot looser, as the intended audience is yourself and whoever is grading your work or advising you. As long as you have some mutually agreed upon convention, it doesn't matter what notation you use. So pick something you like, and stick with it.

In more formal settings (e.g. in a paper which needs to get through peer review), you should expect to adopt whatever notation the editor and/or reviewers recommend.

For this particular notational issue:

• In many contexts, the distinction between a point in some $n$-dimensional space and a more general notion of "point" isn't very important, so it is common to see notation like $$ x = (x_1, x_2, \dotsc, x_n), $$ where no special emphasis is given to the name of the point.

• In more elementary texts, where authors are often trying to use notation to distinguish between different "kinds" of objects, a point in $n$-dimensional space will often be bolded (as noted in the question), i.e. $$ \mathbf{x} = (x_1, x_2, \dotsc, x_n).$$

• When writing notes by hand, the letter can be "decorated" in a number of ways to indicate that the symbol has been "bolded" (or otherwise distinguished). The usual ways of doing this are to either overline or underline the variable of interested, i.e. $$ \overline{x} = \underline{x} = (x_1, x_2, \dotsc, x_n). $$

• Often, the $n$-dimensional space being considered is a vector space, which implies a bit more structure: points in such a space can be added to each other, or combined via an inner product, or scaled (for example). When working with vectors, it is not uncommon to write them (by hand) with a little arrow over the top, e.g. $$ \vec{x} = \langle x_1, x_2, \dotsc, x_n \rangle. $$

• It is also worth noting that a lot of the symbols which are commonly used in mathematics are digitized versions of handwritten approximations of bolded characters. For example, $\mathbb{R}$ is a typeset version of a handwritten bolded letter. Typically, only uppercase letters are given this "blackboard bold" treatment, but you can do it for lowercase letters, too, e.g. $$ \mathbb{x} = (x_1, x_2, \dotsc, x_n). $$ This is not common notation, but if it is for your own use only, who cares?