Interpreting $ F^S$: Functions, Tuples, and Vector Spaces

Question

Currently reading Linear Algebra Done Right by Axler, the author states:

If $S$ is a set, then $F^S$ denotes the set of functions from $S$ to $F$. (...)
Our previous examples of vector spaces, $F^n$ and $F^\infty$,
are special cases of the vector space $F^S$ because a list of length $n$ of numbers in $F$
can be thought of as a function from $\{1,2,\dots,n\}$ to $F$,
and a sequence of numbers in $F$ can be thought of as a function from the set of
positive integers to $F$.

Thus, we can write:
$$ F^n = F^{\{1,2,\dots,n\}}. $$

I have come across related discussions on Math SE:

• What is $\mathbb{R}^n$?
• How is $\mathbb{R}^n$ a function space?

My understanding

Given the definition of function spaces, an element of
$$ F^{\{1,2,\dots,n\}} = \{ f \mid f: \{1,2,\dots,n\} \to \mathbb{R} \} $$ can be represented explicitly as a set of ordered pairs:
$$ \{(1, x_1), (2, x_2), \dots, (n, x_n)\}. $$ Alternatively, the notation used in vector spaces is:
$$ \{(f(1), f(2), \dots, f(n)) \mid f: \{1,2,\dots,n\} \to \mathbb{R} \}. $$ which does not explicitly list input-output pairs.

Question

Is my interpretation of
$$ F^S = \{ f \mid f: \{1,2,\dots,n\} \to \mathbb{R} \} $$ correct, particularly in how I represent elements as sets of ordered pairs? If not, what is the precise way to bridge these notations?

Furthermore, how do I rigorously justify the equivalence:
$$ \{ f \mid f: \{1,2,\dots,n\} \to \mathbb{R} \} = \{(f(1), f(2), \dots, f(n)) \mid f: \{1,2,\dots,n\} \to \mathbb{R} \}. $$

Lastly, are there any recommended materials where I can further explore and practice this concept?

Thanks in advance, sorry if this turns out to be trivial. :)

Answer 1

Other than notational convenience, there is no difference. Writing a vector as a tuple of coordinates $(x_1, \ldots x_n)$ is more convenient than writing it out as $\{(1,x_1), \ldots (n,x_n)\}$. But you can easily and unambiguously move from one notation to the other. Similarly, a sequence $(x_1, x_2, \ldots)$ can also be written as $\{1,x_1), (2,x_2) \ldots \}$. In tuples and sequence, the order of terms matters, but in a set, the order does not matter, and the first element of each ordered pair in the set identifies the ordering to be used when converting it to a tuple or sequence.

If $S$ is uncountable (for example, the set of all real numbers in the unit interval $[0,1]$), the function notation is the only way to write things out, and so this is a more general notation.