Function Application and its Notation

Question

For years and years and years I've always been taught that in mathematics, functions are applied as $f(x)$. But in my university textbook they also use three other notations:

$$f\ x,$$ $$fx,$$ $$\mathcal{F}⟦x⟧.$$

Is there any significance as to how the functions are applied, or more precisely, do these different notations mean anything specific, or are they all just variations of the same thing?

Answer 1

A lot of the time, mathematical notation is decided by historical reasons rather than logical ones. This is true with function application. These all mean the same thing, unless it's obvious from context that they don't.

For example, the notation $fx$ is common when $f$ is a linear transformation. Indeed, $Ax$ is common shorthand for $A(x)$; they mean the exact same thing.

In Aluffi's Algebra - Chapter 0, he discusses the various conventions in which the bijections of the symmetric group are notated.

The various conventions clash in the way the operation in $S_A$ should be written. From the 'automorphism' point of view, if $f, g \in S_A = \text{Aut}_{\text{Set}}(A)$, the 'product' of $f$ and $g$ should be written $g \circ f \dots$ the prevailing style of notation in group theory would write this element as $fg$, apparently reversing the order in which the operation is performed.

Everything would fall back in to agreement if we adopted the convention of writing functions after the elements on which they act rather than before: $(p)f$ rather than $f(p)$. But one cannot change century-old habits, so we have no alternative but to live with both conventions and to state carefully which one we are using at any given time.