What's the meaning of square brackets $[~]$ Below is a list that tries to be exhaustive about the usage of square brackets.  I tried to arrange them so that more common usages come first. Maybe such a list can never be complete, but are there uses other than these below?

*

*Closed Intervals like $[a,b] = \{x\in\Bbb R\mid a\leqslant x\leqslant b\}$.  Open or half-open intervals are usually written with parentheses at the respective end, i.e. $(a,b)$, $(a,b]$ or $[a,b)$. Sometimes reverse brackets are used: $]a,b[$, $]a,b]$ and $[a,b[$ which might have non-matching brackets, a feature that should be avoided, IMHO.


*$R[x]$ or $R[x,y]$ for polynomials in $x$ resp. in $x$ and $y$ with coefficients in $R$. In most cases, $R$ is a ring, an integral domain or even a field, and $R[x]$ and $R[x,y]$ etc. denote Polynomial Rings. Similar notation is $R[[x]]$ to denote (formal) Power Series over $R$.


*$E[X]$ for the Expected Value of a random variable or probability distribution $X$, but notations like $EX$ and $E(X)$ are also used. Frequently  $\Bbb E$ is used instead of $E$. $E[X|Y]$ denotes expectation value for conditional probability. Sometimes completely different notation is used like $\bar X$ or $\overline X$ in physics.


*With sub- and superscript for the difference of respective two function values like in $$\int_a^b\!\! f(x)\,dx = \big[F(x)\big]_{x=a}^{x=b} = F(b)-F(a)$$


*With subscript used to indicate that a (complicated) function or expression is evaluated at that specific point, like in $$\left[\frac{\partial}{\partial x} f(x, \dot x, t)\right]_{t=1}$$


*To denote the Equivalence Class of elements that are equivalent to an element $x$: $$[x] = \{y\mid y\sim x\}$$ where $\sim$ is an equivalence relation.


*To denote the Homogeneous Coordinates of, say, a point in projective space $P\Bbb R^2$ as $[x:y:z]$ or $(x:y:z)$.


*Stirling Numbers of the 1st Kind as $\begin{bmatrix}n\\k\end{bmatrix}$


*For Simple Continued Fractions $$[a_0;a_1,a_2,\ldots] ~=~ a_0+ \underset{i=1}{\overset{\infty}{\Large\text{K}}}\,\frac1{a_i} ~=~ a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}$$


*$[a,b]$ for the Least Common Multiple, similar to the notation $(a,b)$ for the Greatest Common Divisor.


*$f[x]$ for a function that's defined on a discrete set like on $\Bbb Z$ or $\Bbb N$ or for a Time-Discrete Signal, for example in the context of signal analysis and Z-Transform like ${\cal Z}\{x[n]\}$ for the ${\cal Z}$-Transform of time-discrete signal $x$.


*For the Value of a Functional at a specific place like ${\cal F}[f]$ for the Fourier transform of $f$.  More common are notations ${\cal F}\{f\}$, ${\cal F}(f)$ or just ${\cal F}f$.


*$[a,b]$ for the Lie Bracket in a Lie algebra or Lie ring.


*The Lie Bracket of a vector field.  Like the Lie bracket in a Lie algebra it's a binary operation that's bilinear, anti-symmetric and obeys the Jacobi identity $[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0$.


*$[a,b]=ab-ba$ for the Commutator in Group and Ring theory that measures the degree of non-commutativity of an operation. In a group (where there's only one operation) it is $[a,b] = a^{-1}b^{-1}ab$.


*$[X]$ for the Iverson Bracket, a generalization of Kronecker δ. For some expression / predicate $X$ that bracket evaluates to $1$ when $X$ is true, and to $0$ when $X$ is false. Kronecker δ represents as $\delta_{ij}=[i=j]$ for example.


*$[n]_q$ for the q-Analog of $n$, also called q-bracket or q-number.


*As Gauss Bracket $[x]$ to denote the greatest integer not greater than $x$, in programming sometimes called floor function. Iverson's notation $\lfloor x\rfloor$ is clearer and removes ambiguity due to the sheer number of different usages of $[~]$.


*$[a,b,c] = (ab)c-a(bc)$ for the Associator that measures the degree of non-associativity of an operation.


*For matrices and vectors.  Some authors use $\begin{bmatrix} a&b\\c&d\end{bmatrix}$ instead of $\begin{pmatrix} a&b\\c&d\end{pmatrix}$ and $\begin{bmatrix} x\\y\end{bmatrix}$ instead of $\begin{pmatrix} x\\y\end{pmatrix}$ etc.


*Instead of parentheses $()$ in order to to "override" the conventions for Precedence of Operations and operators and to determine in which order to evaluate an expression, like in $p(x)=x[1 + x(1+x)]$ instead of $p(x)=x(1 + x(1+x))$.  Sometimes even mixed with braces $\{\}$ to add more confusion.

I am not really sure about the "Matrices and Vectors" point and that they really mean the same.  So is that just an author's preference, some typographic consideration or even different semantics?
 A: 

*Let $X$ be a linear space (or vector space) and $x, y\in X$
Then $[x, y]=\{(1-t)x+ty:t\in [0,1]\}$ is the line segment joining $x$ and $y$.
$ S\subset X$ is convex if $\quad \forall x, y\in S$ ,$[x, y]\subset S$
Let $S\subset X_{\Bbb{R}}$ .Then $f:S\subset \Bbb{R}^d\to \Bbb{R}$ is convex function if $\forall x, y\in S, t\in [0,1]$ $$f([x, y]_t)\le [f(x), f(y) ]_t$$



*(Generalization of $1$)

$(X, \le )$ be a linear ordered set.
Then $[x, y]=\{z\in X : x\le z\le y\}$ is called a closed interval in $X$. This notion of "closeness" is consistent with the "closed set" in the topology (order topology) on $X$



*Let $G$ be a group and $H\le G$.

Then the number of distinct costs (left/right) is called the index of $H$ in $G$ and it is denoted by $[G:H]$
$[$ Alternative notation : $i_G(H) \quad]$



*Let $L\mid_{K} $ be a field  extension.

Then $[L:K]=\dim (L_K) $ denote the degree of the field extension $L$ over $K$.



*Let $L\mid_{K} $ be a field  extension and $A\subset L$.

$$K[A]=\bigcap_{K,A\subset S} S$$  where $S\subset L$ is subring of $L$.
In other words $K[A]$ is the smallest subring of $L$ containing $K$ and $K$.
$K[A]$ is called ring adjunction of $A$ to $K$.
On the other hand $K(A)$ ( smallest sub-field of $L$ containing both $K$ and $A$ ) is called field adjunction of $A$ to $K$

A: This answer uses
$$
\color{red}[z^{31}\color{red}] \frac{z}{1 - z} 
           \cdot \frac{1}{1 - z^2}
           \cdot \frac{z}{1 - z^2}
           \cdot \frac{1}{(1 - z)^2}
$$
to mean the coefficient of $z^{31}$ in the expansion of the above as a power series.
