The Summation Map on Real-Valued Maps over a Finite Set, and the Dot Product, Euclidean Norm and Euclidean Distance as Maps

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces, with maps and their values as in Sets and Maps: Ordinary Notation §maps.

¶ Let $B$ be a finite set, $\mathbb{R}^{B}$ the set of maps from $B$ to $\mathbb{R}$, and sums over $B$ as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings. There is exactly one map $\Sigma_{B}:\mathbb{R}^{B}\to\mathbb{R}$ with $\Sigma_{B}(a)=\sum_{k\in B}a(k)$ for every $a\in\mathbb{R}^{B}$.

In the clauses dot-product, norm and distance, let $n\in\mathbb{N}$, with the difference of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §operations, the dot product and the Euclidean norm $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$.

¶ There is exactly one map $f:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ with $f(x,y)=x\cdot y$ for all $x,y\in\mathbb{R}^{n}$.

¶ There is exactly one map $g:\mathbb{R}^{n}\to\mathbb{R}$ with $g(x)=\lVert x\rVert$ for every $x\in\mathbb{R}^{n}$.

¶ There is exactly one map $h:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$ with $h(x,y)=\lVert x-y\rVert$ for all $x,y\in\mathbb{R}^{n}$.