Common Symbols

$P \implies Q$

$P$ implies $Q$

$P \iff Q$

$P \implies Q$ and $Q \implies P$

$\natset{n}$

the set $\{1, \ldots, n\}$

$\alpha := 1$

$\alpha$ is defined as equal to $1$

$f \equiv 1$

function $f$ is everywhere equal to $1$

$\wp(A)$

the power set of $A$; that is, the collection of all subsets of set $A$

$\NN$, $\ZZ$ and $\RR$

the natural numbers, integers and real numbers respectively

$\CC$

the set of complex numbers (see Section 6.1.1.8)

$\ZZ_+$, $\RR_+$, etc.

the nonnegative elements of $\ZZ$, $\RR$, etc.

$\matset{n}{k}$

all $n \times k$ matrices

$\diag(a_1, \ldots, a_n)$

the diagonal matrix with $a_1, \ldots a_n$ on the principle diagonal

$|x|$

the absolute value of $x \in \RR$

$|B|$

the cardinality of (number of elements in) set $B$

$f \colon A \to B$

$f$ is a function from set $A$ to set $B$

$x \mapsto y$

$x$ is mapped to $y$ (under some given function)

$\RR^S$

the set of all functions from $S$ to $\RR$

$\RR^n$

all $n$-tuples of real numbers

$\| x \|_1$

the $\ell_1$ norm $\sum_i |x_i|$ (see Section 2.3.1.1)

$\| x \|_\infty$

the $\ell_\infty$ norm $\max_i |x_i|$ (see Section 2.3.1.1)

$\| A \|$ when $A \in \matset{n}{k}$

the operator norm of $A$ (see Section 2.3.2.3)

$\la a, b \ra$

the inner product of $a$ and $b$

$a \vee b$

the maximum of $a$ and $b$ (pointwise if they are vectors)

$a \wedge b$

the minimum of $a$ and $b$ (pointwise if they are vectors)

$g \ll h$

function (or vector) $g$ is everwhere strictly less than $h$