What's the meaning of square brackets $[~]$

Question

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?

1. 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.

2. $R[x]$ 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 $R$.

3. $E[X]$ 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.

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

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

6. 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.

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

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

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

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

11. $f[x]$ 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 $x$.

12. 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 ${\cal F}f$.

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

14. 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 $[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0$.

15. $[a,b]=ab-ba$ 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 $[a,b] = a^{-1}b^{-1}ab$.

16. $[X]$ 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.

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

18. 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 $[~]$.

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

20. 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.

21. 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.

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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?

Answer 1

22. 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 $\forall x, y\in S, t\in [0,1]$ $$f([x, y]_t)\le [f(x), f(y) ]_t$$

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23. (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 $X$

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24. 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 : $i_G(H) \quad]$

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

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

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26. 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 $K$

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Answer 2

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.

Answer 3

• If $A$ is an abelian group and $n \in \mathbb{Z}$, then $A[n]$ sometimes denotes the kernel of $n :A \longrightarrow A$, that is, the subgroup $\{a \in A : n \cdot a = 0\}$.
• If $R$ is a graded ring and $M$ is a graded $R$-module, and $n \in \mathbb{Z}$, then $M[n]$ denotes the graded $R$-module which is shifted from $M$ by $n$ steps. Formally, $M[n]_i := M_{n+i}$.
• If $C$ is a chain complex of abelian groups (or in any abelian category), then $C[1]$ denotes the chain complex which is shifted by one (again, $C[1]_i := C_{i+1}$) and whose differentials get a minus sign. Similarly for $C[n]$, where $n \in \mathbb{Z}$. The notation is also used more generally in triangulated categories.
• If $R$ is a ring and $M$ is a monoid, then $R[M]$ denotes the monoid algebra of formal linear combinations $\sum_m r_m \cdot m$, $r_m \in R$. The already mentioned polynomial ring is a special case of this construction (take a free commutative monoid). When $M = \mathbb{Z}$, we get $R[X,X^{-1}]$, the ring of finite Laurent series over $R$.
• When $R$ is a ring and $M$ is some $R$-module, some authors write $R[M]$ for the $R$-algebra whose underlying $R$-module is $R \oplus M$, in which the unit is $1 \in R$ and multiplication is characterized by $M \cdot M = 0$. (Thus, $(r+m)(s+n) = rs + (rn+sm)$.) For example, $R[R] \cong R[\varepsilon]/\langle \varepsilon^2 \rangle$ will be the algebra of dual numbers over $R$.