Notation#

This section summarizes major notation used in the book.

Numbers#

Notation

Meaning

Reference

\(\Nat\)

The set of natural numbers

\(\ZZ\)

The set of integers

\(\QQ\)

The set of rational numbers

\(\RR\)

The set of real numbers

Definition 2.1

\(\ERL\)

The extended real line \([-\infty, \infty]\)

Definition 2.40

\(\CC\)

The set of complex numbers

\(\Re(x)\)

The real part of a complex number

\(\Im(x)\)

The imaginary part of a complex number

Sets and Functions#

Notation

Meaning

Reference

\(\dom f\)

Domain of a function \(f\)

Definition 1.49

\(\range f\)

Range of a function \(f\)

Definition 1.51

\(\epi f\)

Epigraph of a function \(f\)

\(\supp f\)

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

\(\VV\)

A vector space (usually finite dimensional)

\(\EE\)

A normed vector space (usually finite dimensional and Euclidean)

\(\RR^n\)

\(n\) dimensional Euclidean real vector space

\(\RR^{m \times n}\)

The space of \(m \times n\) real matrices

\(\SS^{n}\)

The space of \(n \times n\) symmetric real matrices

\(\NullSpace(A)\)

Null space of a matrix \(A\)

\(\ColSpace(A)\)

Column space of a matrix \(A\)

\(\RowSpace(A)\)

Row space of a matrix \(A\)

\(\Range(A)\)

Range of a set of vectors \(A\)

\(\Nullity(A)\)

Nullity of a an operator \(A\)

\(\Trace(A)\)

Trace of a matrix \(A\)

\(\Diag(A)\)

Diagonal of a matrix \(A\)

\(\supp(v)\)

Support of a vector \(v\) (non-zero indices)

\(\bzero\)

The all zeros vector

\(\bone\)

The all ones vector

Topology / Metric Spaces#

Notation

Meaning

\(\interior A\)

The interior of a set \(A\)

\(\closure A\)

The closure of a set \(A\)

\(\boundary A\)

The boundary of a set \(A\)

\(\diam A\)

The diam of a set \(A\)

\(\relint A\)

The relative interior of a set \(A\)

Calculus#

Notation

Meaning

Reference

\(\lim_{x \to a} f(x)\)

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

\(f'\)

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

\(f'_-(a)\)

Left hand derivative of \(f\) at \(x=a\)

Definition 2.83

\(f'_+(a)\)

Right hand derivative of \(f\) at \(x=a\)

Definition 2.83

\(\nabla f\)

Gradient of \(f\)

Convex Analysis#

Notation

Meaning

\(\prox_f\)

The proximal operator for a function \(f\)