Component Maps and Entry Maps

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $p,q\in\mathbb{N}$ and let $S$ be a set. For $z\in\mathbb{R}^{q}$ and $k\in[q]$, $z_{k}$ is the $k$-th component of $z$, and for $B\in\mathcal{M}_{p\times q}(\mathbb{R})$, $i\in[p]$ and $j\in[q]$, $B_{ij}$ is the entry of $B$ in row $i$ and column $j$, as in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (iii).

(i) Let $x:S\to\mathbb{R}^{q}$ be a map and let $k\in[q]$. The $k$-th component map of $x$ is the map $x_{k}:S\to\mathbb{R}$ with $x_{k}(s)=x(s)_{k}$ for every $s\in S$; it exists and is unique by Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (i), and it is an element of the set $\mathbb{R}^{S}$ of maps from $S$ to $\mathbb{R}$ by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i).

(ii) Let $A:S\to\mathcal{M}_{p\times q}(\mathbb{R})$ be a map, and let $i\in[p]$ and $j\in[q]$. The entry map of $A$ in row $i$ and column $j$ is the map $A_{ij}:S\to\mathbb{R}$ with $A_{ij}(s)=A(s)_{ij}$ for every $s\in S$; it exists and is unique by Component Maps of a Map into $\mathbb{R}^{q}$, Entry Maps of a Map into Matrices, and Maps Assembled from Them (ii), and it is an element of $\mathbb{R}^{S}$ by The Elements of the Class of Functions from a Set to a Set, and This Class Is a Set (i).