I have a question regarding the syntax for multivariate functions and corresponding definitions. My understanding so far is a function accepts a single argument and this can (should?) be a multi-argument list passed as an ordered tuple.
If:
$f: X \rightarrow Y_0 \times Y_1$
$g: Y_0 \times Y_1 \rightarrow Z$
and an ordered tuple is the result of $f$ and an argument to $g$.
Commonly defined:
$(g \circ f)(x) = g(f(x))$
Should this definition really be:
$(g \circ f)x = gfx$
omitting the parenthesis because the result of $f$ is an ordered tuple? I.e. already is enclosed in parenthesis.
Confusing the matter more, Wiki says that a
function of n variables is a function whose domain is a set of n-tuples
and says for a function:
$f : X_1 \times … \times X_n \rightarrow Y$
When using function notation, one usually omits the parentheses surrounding tuples,
So when writing the form $f((x_1, …, x_n))$, the parenthesis is omitted and instead we write $f(x_1, …, x_n)$.
I'm a bit confused and could use guidance. For example, I see Wiki define $(f + g)(x) = f(x) + g(x)$ and am asking myself, does this definition work with multi-variate functions. Again, I would expect this definition to be $(f + g)x = fx + gx$ for a multi-variate argument. I recall seeing some older mathematics literature and saw use with parenthesis omitted. For this reason, I'm expanding to math-history.
Best Answer
The parentheses for an element of a product space do not have the semantics of the parenthesis for evaluation of a function. The way you are writing expressions suggests you interpret $$ f(x) $$ as "$f$" and "$(x)$", where the mere mention of $f$ means to use it for evaluation and the parenthesized expression is a $1$-tuple. This is not the semantics of those uses of those symbols. $f$ is the name/label of a function, the parentheses have the semantics of evaluation -- in particular, evaluate the function named prior to the parentheses on the input enclosed in the parentheses.
The expression $fx$ is not normally a valid sentence since it is two adjacent nouns. If you intend to evaluate $f$ at $x$, parentheses are needed.
Consequently, in $$ f((x_1, x_2, \dots, x_n)) $$ the outer parentheses have the semantics of function evaluation and the inner parentheses have the semantics of $n$-tuple aggregation. It is a lexicographical hack (to improve legibility) to combine the doubled open parentheses into a single open parenthesis and the doubled close parentheses into a single close parenthesis. This combination is merely one of presentation in textual form, it represents the doubled form, with both evaluation and aggregation parentheses.