Common Symbols and Terminology - Dynamic Programming Volume I: Finite States

The following symbols and conventions are used throughout the book.

Symbol	Meaning
$\1\{P\}$	indicator, equal to 1 if statement $P$ is true and 0 otherwise
$\alpha \coloneqq 1$	$\alpha$ is defined to be equal to $1$
$f \equiv 1$	function $f$ is everywhere equal to $1$
$\wp(A)$	the power set of $A$ — the collection of all subsets of $A$
$\natset{n}$	$\{1, \ldots, n\}$
$\NN$ , $\ZZ$ , $\RR$ and $\CC$	the natural, integer, real, and complex numbers respectively
$\ZZ_+$ , $\RR_+$ , etc.	the nonnegative elements of $\ZZ$ , $\RR$ , etc.
$\lvert x\rvert$ for $x \in \RR$	the absolute value of $x$
$\lvert \lambda\rvert$ for $\lambda \in \CC$	the modulus of $\lambda$ (i.e., $\sqrt{a^2+b^2}$ if $\lambda=a+ib$ )
$\lvert B\rvert$ for set $B$	the cardinality of $B$
$\RR^n$	all $n$ -tuples of real numbers
$x \leq y$ for $x,y \in \RR^n$	$x_i \leq y_i$ for $i=1,\ldots,n$ (pointwise partial order)
$x \ll y$ for $x,y \in \RR^n$	$x_i < y_i$ for $i=1,\ldots,n$
$\dD(F)$	the set of distributions on $F$
$\RR^{\Msf}$	the set of all functions from $\Msf$ to $\RR$
$i\RR^{\Msf}$	the set of increasing functions in $\RR^{\Msf}$
$\lopx$	the set of linear operators on $\RR^{\Xsf}$
$\mopx$	the set of Markov operators in $\lopx$ (see Section 2.3.3.4)
$\la a, b \ra$	inner product of the vectors $a$ and $b$
$\bigvee_{\alpha \in A} u_\alpha$	the supremum of $\{u_\alpha\}_{\alpha \in A}$
$\bigwedge_{\alpha \in A} u_\alpha$	the infimum of $\{u_\alpha\}_{\alpha \in A}$
IID	independent and identically distributed
$X \eqdist Y$	$X$ and $Y$ have the same distribution
$X \sim F$	$X$ has distribution $F$
$F \lefsd G$	$G$ first-order stochastically dominates $F$