# Function notation/mapping for this type of functions?

In short, what is the proper function notation/mapping for this type of functions?

From Wikipedia regarding maps:

In the communities surrounding programming languages that treat functions as first-class citizens, a map is often referred to as the binary higher-order function that takes a function $$f$$ and a list $$[v_0, v_1, \dots, v_n]$$ as arguments and returns $$[f(v_0), f(v_1), ..., f(v_n)]$$ (where $$n \geq 0$$).

In mathematics we use the function and mapping notation \begin{align} f&:\mathbb R^n\rightarrow \mathbb R^n \tag 1 \\ x&\mapsto (f_1(x), \dots, f_n(x)) \tag 2 \end{align} Where $$x=(x_1,\dots, x_n)$$ and we often write $$f(x_1,\dots, x_n)=( f_1(x_1,\dots, x_n), \dots, f_n(x_1,\dots, x_n) ) \tag 3$$

Question:

According Wikipedia we have the variable $$v=(v_0,\dots, v_n)$$ and a function of the form \begin{align} f(v_0,\dots, v_n)= (f(v_1), f(v_2), \dots, f(v_n)) \tag 4 \end{align} But this form doesn't correlate with the notation in $$(1)-(3)$$. What is the proper function/mapping notation for $$(4)$$?

• What is the question precisely? Whether there is a standard mathematical notation for what computer scientists call map? – Andrej Bauer Dec 4 '20 at 16:43
• The (computer science) mapping of a function on the components of a tuple corresponds to the Cartesian product (Cartesian power in this case) of $f$ with itself. If you have, for example $f:\mathbb{R}\to\mathbb{R}$, then you have $\prod_{i=1}^{n}f:\mathbb{R}^n\to\mathbb{R}^n$ defined as $(\prod_{i=1}^{n}f)(v_1,v_2,...,v_n)=(f(v_1),f(v_2),...,f(v_n))$. This notation is not used enough to use it without explicitly defining it. – plop Dec 4 '20 at 17:34
• For example, the Wikipedia article is denoting it as $\langle f,f,...,f\rangle$, which is also not widely used either. – plop Dec 4 '20 at 17:39
• @plop Can you explain your notation further? You have the product symbol over $i$ but no $i$ in your function, is it a typo? Do you mean $\prod_{i=1}^{n}f(v_i)=(f(v_1),f(v_2),...,f(v_n))$? – JDoeDoe Dec 8 '20 at 15:33
• There is no missing $i$. If the $i$ is not in the factor, is because the factor is independent of $i$ and it is being multiplied (Cartesian product) with itself $n$ times. – plop Dec 8 '20 at 15:44