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I'm wondering if there is any definitive style guide for writing mathematics. In particular, I'm looking for rules for when to use '$( \ )$' versus '$[ \ ]$.' For instance, when referring to a function, most texts will write $f(x)$, not $f[x]$. On the other hand, when referring to the expectation of a random variable (which I guess is a function too) most (but not all) texts write $\mathbf{E}[X]$, not $\mathbf{E}(X).$

Is there a reason for this or is it a matter of the personal preference of the author? Are there places where it would be 'wrong' to replace '$( \ )$' with '$[ \ ]$' and vice versa (ignoring cases like interval notation where the mathematical meaning is different)?

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    It's sort of a long bit of tradition and formalism that one gets used to. When I see $F[x]$, I think of a functional. When I see $f(x)$, I think of a function. Brackets for Field Extensions, parenthesis for ideals or generators. So lots of cases where they aren't interchangeable. But there is little wrong with writing $E(X)$, I would say.2011-09-16
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    I realized after I wrote this that the wikipedia page for expected value use both notations, so maybe I made too bold a statement when I said 'most' texts write $\mathbf{E}[X]$.2011-09-16
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    I don't believe there are any specific rules. The convention I sometimes use is to reserve square brackets for higher-order functions. For example, a random variable is a function on sample space, so the expectation operator is a higher-order function.2011-09-16
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    $E[X]$ is a functional; its input is a probability distribution and it outputs a real number.2011-09-16
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    This is helpful. It turns out that most of the examples I had in mind where brackets are used instead of parentheses involved functionals--I had just not noticed what they had in common.2011-09-16
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    The notation $f[S]$ is sometimes used for the set $\{f(s):s\in S\}$, although $f(S)$ is probably more common for this except when confusion can arise.2011-09-16
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    @Samuel To make matters more interesting, I've seen $f^*(S)$ and $f\left\langle S\right\rangle$ used to refer to that same set.2015-04-24

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