Math notation to define the operator that extract component of a vector?

Question

Say $\mathbf{x} \in \mathbb{R}^n$, what is the common notation to extract the first component as an operation?

Something like $\mathcal{P}_j = \mathbf{x}_j$? $\mathbf{x}_j$ is the j-th component.

I need this because I have a function $f(\mathbf{x}) \approx \mathbf{x}$, so I need something like $\mathcal{P}f(\mathbf{x}) \approx \mathbf{x}_j$ since my metric is defined for a single component of $\mathbf{x}_j$.

Answer 1

The Wikipedia article Projection suggests $\textrm{proj}_j(x)$ as in this excerpt

An operation typified by the $\ j\ $th projection map, written $\ \textrm{proj}_j\ $, that takes an element $\ x = (x_1, \dots, x_j , \dots, x_k)\ $ of the cartesian product $X_1 \times \dots \times X_j \times \dots \times X_k\ $ to the value $\ \textrm{proj}_j (x) = x_j.\ $ This map is always surjective.

I think this particular notation is rarely used. However, a more common notation is $\ \pi_j\ $ as in this excerpt from the Wikipedia article Product

An object $\ X\ $ is the product of a family $\ (X_i)_{\ i\in I}\ $ of objects iff there exist morphisms $\ \pi_i : X \to X_i\ $, such that for every object $\ Y$

I have seen $\ \pi_j\ $ or the variant $\ p_j\ $ used more often and these are closer to a standard notation, but as the projection Wikipedia article shows it is not always used. As in so many areas of usage of language the process is not rational or standardized. It depends on social processses and chance.

Whatever notation you decide on, you should explain it the first time you use it.