Notation

This section summarizes major notation used in the book.

Numbers

Notation	Meaning	Reference
$\mathbb{N}$	The set of natural numbers
$\mathbb{Z}$	The set of integers
$\mathbb{Q}$	The set of rational numbers
$\mathbb{R}$	The set of real numbers	Definition 2.1
$\bar{\mathbb{R}}$	The extended real line $[-\mathrm{\infty },\mathrm{\infty }]$	Definition 2.40
$\mathbb{C}$	The set of complex numbers
$\mathrm{Re}(x)$	The real part of a complex number
$\mathrm{Im}(x)$	The imaginary part of a complex number

Sets and Functions

Notation	Meaning	Reference
$\mathrm{dom}f$	Domain of a function $f$	Definition 1.49
$\mathrm{range}f$	Range of a function $f$	Definition 1.51
$\mathrm{epi}f$	Epigraph of a function $f$
$\mathrm{supp}f$	Support of a function $f$
$g\circ f$	Composition of functions $g$ and $f$ with $(g\circ f)(x)=g(f(x))$	Definition 1.58

Linear Algebra

Notation	Meaning
$\mathbb{V}$	A vector space (usually finite dimensional)
$\mathbb{E}$	A normed vector space (usually finite dimensional and Euclidean)
${\mathbb{R}}^n$	$n$ dimensional Euclidean real vector space
${\mathbb{R}}^{m\times n}$	The space of $m\times n$ real matrices
${\mathbb{S}}^n$	The space of $n\times n$ symmetric real matrices
$\mathcal{N}(A)$	Null space of a matrix $A$
$\mathcal{C}(A)$	Column space of a matrix $A$
$\mathcal{R}(A)$	Row space of a matrix $A$
$\mathrm{R}(A)$	Range of a set of vectors $A$
$\mathrm{nullity}(A)$	Nullity of a an operator $A$
$\mathrm{tr}(A)$	Trace of a matrix $A$
$\mathrm{diag}(A)$	Diagonal of a matrix $A$
$\mathrm{supp}(v)$	Support of a vector $v$ (non-zero indices)
$\mathbf{0}$	The all zeros vector
$\mathbf{1}$	The all ones vector

Topology / Metric Spaces

Notation	Meaning
$\mathrm{int}A$	The interior of a set $A$
$\mathrm{cl}A$	The closure of a set $A$
$\mathrm{bd}A$	The boundary of a set $A$
$\mathrm{diam}A$	The diam of a set $A$
$\mathrm{ri}A$	The relative interior of a set $A$

Calculus

Notation	Meaning	Reference
$\underset{x\to a}{lim}f(x)$	Limit of $f$ as $x$ approaches $a$	Definition 2.64
$x\to a^{-}$	$x$ approaches $a$ from the left	Definition 2.65
$x\to a^{+}$	$x$ approaches $a$ from the right	Definition 2.65
$f(a^{-})$	Left hand limit of $f$ at $x=a$	Definition 2.65
$f(a^{+})$	Right hand limit of $f$ at $x=a$	Definition 2.65
$f^{\prime }$	First derivative of $f$	Definition 2.78
$f^{(1)}$	1st derivative of $f$	Definition 2.81
$f^{(n)}$	n-th derivative of $f$	Definition 2.81
$f^{(0)}$	0-th derivative of $f$ ($f^{(0)}=f$)	Definition 2.81
$f_{-}^{\prime }(a)$	Left hand derivative of $f$ at $x=a$	Definition 2.83
$f_{+}^{\prime }(a)$	Right hand derivative of $f$ at $x=a$	Definition 2.83
$\mathrm{\nabla }f$	Gradient of $f$

Convex Analysis

Notation	Meaning
${\mathrm{prox}}_f$	The proximal operator for a function $f$