## Index Index

Abel’s theorem, Exercise

absolute convergence, Definition

absolute value, Paragraph

achieves absolute maximum, Paragraph

achieves absolute minimum, Paragraph

additive property of the integral, Paragraph

algebra, Paragraph Definition

algebraic geometry, Example

algebraic number, Exercise

almost every, Paragraph

analytic, Paragraph

analytic function, Paragraph

antiderivative, Paragraph

approximate delta function, Subsection Paragraph

arc-length measure, Paragraph

arc-length parametrization, Paragraph

Archimedean property, Item

arithmetic-geometric mean inequality, Exercise

Arzelà--Ascoli theorem, Theorem

axiom, Paragraph Definition

ball, Definition

Banach fixed point theorem, Theorem

base of the natural logarithm, Paragraph

basis, Definition

Bernoulli’s inequality, Exercise

Bessel’s inequality, Paragraph

bijection, Definition

bijective, Definition

bilinear, Paragraph

binary relation, Definition

bisection method, Paragraph

Bolzano--Weierstrass theorem, Theorem

Bolzano–Weierstrass theorem, Exercise

Bolzano’s intermediate value theorem, Theorem

Bolzano’s theorem, Theorem

boundary, Definition

bounded above, Item

sequence, Definition

bounded below, Item

sequence, Definition

bounded domain with piecewise smooth boundary, Definition

bounded function, Definition Paragraph

bounded interval, Paragraph

bounded sequence, Definition Definition

bounded set, Definition Definition

bounded variation, Item

Cantor diagonalization, Paragraph

Cantor function, Exercise

Cantor set, Example

Cantor–Bernstein–Schröder, Paragraph

cardinality, Definition

Cartesian product, Definition

Cauchy

complex series, Paragraph

Cauchy condensation principle, Exercise

Cauchy in the uniform norm, Definition

Cauchy principal value, Paragraph

Cauchy product, Paragraph

Cauchy sequence, Definition Definition

Cauchy series, Definition

Cauchy-complete, Remark Definition

Cauchy–Bunyakovsky–Schwarz inequality, Lemma

Cauchy’s mean value theorem, Theorem

Cesàro summability, Paragraph

chain rule, Proposition Theorem

change of basis, Paragraph

change of variables theorem, Theorem

characteristic function, Paragraph

clopen, Definition

closed ball, Definition

closed interval, Paragraph

closed path, Definition

closed rectangle, Definition

closed set, Definition

closure, Definition

cluster point, Exercise Definition Definition

in a metric space, Definition

codomain, Definition

column, Paragraph

column vectors, Paragraph

commutative diagram, Paragraph

compact, Definition

compact operator, Paragraph

compact support, Paragraph

comparison test for improper integrals, Proposition

comparison test for series, Proposition

complement, Item

complement relative to, Item

complete, Remark Definition

completeness property, Paragraph

complex algebra, Definition

complex conjugate, Example Definition

complex modulus, Example

complex number, Paragraph

complex numbers, Example

complex plane, Paragraph

composition of functions, Definition

conditional convergence, Definition

connected, Definition

conservative vector field, Paragraph

constant sequence, Paragraph

continuous at \(c\), Definition Definition

continuous function, Definition

in a metric space, Definition

continuous function of two variables, Definition

continuously differentiable, Paragraph Definition Definition

continuously differentiable path, Definition

contraction, Definition

contraction mapping principle, Theorem

convergent

improper integral, Definition

power series, Paragraph

sequence, Definition

sequence in a metric space, Definition

series, Definition

converges

complex series, Paragraph

function, Definition Definition

function in a metric space, Definition

power series, Paragraph

converges absolutely, Definition

complex series, Paragraph

converges conditionally, Definition

converges in uniform norm, Paragraph

converges pointwise, Definition Paragraph

complex series, Paragraph

converges to infinity, Paragraph

converges uniformly, Definition Paragraph

converges uniformly on compact subsets, Exercise

convex combination, Paragraph

convex hull, Paragraph

convolution, Item Subsection

cosine, Paragraph

countable, Definition

countably infinite, Definition

curve, Paragraph

Darboux integral, Definition Definition

Darboux sum, Definition Definition

Darboux’s theorem, Theorem

decimal digit, Paragraph

decimal representation, Paragraph

decreasing, Definition Paragraph

Dedekind completeness property, Paragraph

Dedekind cut, Item

DeMorgan’s theorem, Theorem

dense, Exercise

density of rational numbers, Item

derivative, Definition Definition

complex-valued function, Paragraph

determinant, Paragraph

Devil’s staircase, Exercise

diagonal matrix, Item

diagonalization, Paragraph

diameter, Paragraph

difference quotient, Definition

differentiable, Definition Definition

continuously, Paragraph

differentiable curve, Paragraph

\(n\) times, Paragraph

differential equation, Paragraph

differential one-form, Definition

digit, Paragraph

dimension, Definition

Dini’s theorem, Exercise

direct image, Definition

directional derivative, Paragraph

Dirichlet kernel, Paragraph

disconnected, Definition

discontinuity, Paragraph

discontinuous, Paragraph

discrete metric, Example

disjoint, Item

distance function, Definition

divergent

improper integral, Definition

power series, Paragraph

sequence, Definition

sequence in a metric space, Definition

series, Definition

diverges, Definition

function in a metric space, Definition

diverges to infinity, Definition Definition

diverges to minus infinity, Definition

domain, Definition

dot product, Paragraph

eigenvalue, Paragraph

element, Definition

elementary matrix, Paragraph

elementary step function, Paragraph

empty set, Definition

equal, Item

equicontinuous, Paragraph

equivalence class, Definition

equivalence relation, Definition

euclidean norm, Paragraph

euclidean space, Paragraph

Euler–Mascheroni constant, Item

Euler’s formula, Item

Euler’s number, Paragraph

even function, Exercise

even permutation, Paragraph

exponential, Subsection Paragraph

extended real numbers, Paragraph

extreme value theorem, Theorem

field, Definition

finite, Definition

finitely many discontinuities, Paragraph

first derivative, Paragraph

first derivative test, Paragraph

first order ordinary differential equation, Paragraph

fixed point, Definition

fixed point theorem, Theorem

for almost every, Paragraph

Fourier coefficients, Paragraph

Fourier series, Paragraph

Fourier sine and cosine transforms, Exercise

function, Paragraph

bounded, Paragraph

continuous, Definition Definition

differentiable, Definition

Lipschitz, Definition Example

fundamental theorem of algebra, Theorem

fundamental theorem of calculus, Paragraph

general linear group, Proposition

generate an algebra, Paragraph

geometric series, Proposition Paragraph

gradient, Paragraph

graph, Definition

great circle distance, Example

greatest lower bound, Item

Green’s theorem, Theorem

half-open interval, Paragraph

harmonic series, Example

Hausdorff metric, Exercise

Heine--Borel theorem, Theorem

hyperbolic cosine, Paragraph

hyperbolic sine, Paragraph

identity, Definition

identity of indiscernibles, Item

identity theorem, Theorem

image, Definition

imaginary axis, Paragraph

imaginary part, Definition

implicit function theorem, Theorem

improper integrals, Paragraph

increasing, Definition Paragraph

indicator function, Paragraph

induction, Paragraph

induction hypothesis, Paragraph

infimum, Item

infinite, Definition

infinite limit

of a function, Definition

of a sequence, Definition

infinity norm, Definition

initial condition, Paragraph

injection, Definition

injective, Definition

inner product, Paragraph

integers, Item

integrable, Definition

integrable on \(S\), Definition

integral test for series, Proposition

integration by parts, Exercise

interior, Definition

intermediate value theorem, Theorem

intersection, Item

interval, Paragraph

inverse function, Paragraph

inverse image, Definition

invertible linear transformation, Definition

irrational, Paragraph

isolated singularity, Paragraph

Jacobian, Definition

Jacobian conjecture, Paragraph

Jacobian determinant, Definition

Jacobian matrix, Definition

joint limit, Paragraph

Jordan measurable, Paragraph

\(k\)-times continuously differentiable function, Definition

Kronecker density theorem, Exercise

\(L^1\)-convergence, Paragraph

\(L^1\)-norm, Paragraph

Lagrange form, Paragraph

Laplace equation, Example

Laplace transform, Exercise

least element, Paragraph

least upper bound, Item

least-upper-bound property, Definition

Lebesgue covering lemma, Lemma

Lebesgue number, Lemma

Lebesgue--Vitali theorem, Theorem

Leibniz integral rule, Theorem

Leibniz rule, Paragraph

length, Definition

length of a curve, Definition

liminf, Paragraph Definition

limit

infinite, Definition Definition

of a function, Definition

of a function at infinity, Definition

of a function in a metric space, Definition

of a sequence in a metric space, Definition

limit comparison test, Exercise

limit inferior, Paragraph Definition

limit superior, Paragraph Definition

limsup, Paragraph Definition

linear, Definition

linear combination, Definition

linear first order differential equations, Exercise

linear operator, Definition

linear transformation, Definition

linearity of series, Proposition

linearity of the derivative, Proposition

linearity of the integral, Proposition Proposition

linearly dependent, Definition

linearly independent, Definition

Lipschitz continuous, Definition

in a metric space, Example

logarithm, Subsection Paragraph

logarithm base \(b\), Item

longest side, Paragraph

lower bound, Item

lower Darboux integral, Definition Definition

lower Darboux sum, Definition Definition

matrix, Paragraph

maximum, Paragraph

absolute, Paragraph

relative, Definition

strict relative, Paragraph

maximum modulus principle, Theorem

maximum principle

analytic functions, Theorem

harmonic functions, Exercise

maximum-minimum theorem, Theorem

mean value property, Example

mean value theorem for integrals, Exercise

measure zero, Definition

member, Definition

Mertens’ theorem, Theorem

metric, Definition

metric space, Definition

minimum, Paragraph

absolute, Paragraph

relative, Definition

strict relative, Paragraph

minimum modulus principle, Paragraph

minimum-maximum theorem, Theorem

modulus, Example Definition

monic polynomial, Subsection Exercise

monotone convergence theorem, Theorem

monotone decreasing sequence, Definition

monotone function, Definition

monotone increasing sequence, Definition

monotone sequence, Definition

monotonic sequence, Definition

monotonicity of the integral, Proposition Proposition

\(n\) times differentiable, Paragraph

\(n\)-dimensional volume

Jordan measurable set, Paragraph

rectangles, Definition

natural logarithm, Paragraph

natural numbers, Item

naïve set theory, Paragraph

negative, Definition

negatively oriented, Definition

neighborhood, Definition

nondecreasing, Paragraph

nonincreasing, Paragraph

nonnegative, Definition

nonnegativity of a metric, Item

nonpositive, Definition

norm, Definition

normed vector space, Definition

\(n\)th derivative, Paragraph

\(n\)th derivative test, Exercise

\(n\)th order Taylor polynomial, Definition

null set, Definition

nullspace, Paragraph

odd function, Exercise

odd permutation, Paragraph

one-form, Definition

one-sided limit, Paragraph

one-to-one, Definition

onto, Definition

open ball, Definition

open cover, Paragraph

open interval, Paragraph

open mapping, Corollary

open neighborhood, Definition

open rectangle, Definition

open set, Definition

operator norm, Definition

operator, linear, Definition

ordered field, Definition

ordered set, Definition

ordinary differential equation, Paragraph

orthogonal system, Definition

orthonormal system, Definition

oscillation, Paragraph

outer measure, Definition

p-series@\(p\)-series, Proposition

p-test for integrals@\(p\)-test for integrals, Proposition

p-test@\(p\)-test, Proposition

Parseval’s theorem, Theorem

partial derivative, Definition

partial derivative of order \(k\), Paragraph

partial sums, Definition

partition, Definition Definition

path connected, Definition

path independent, Paragraph

Peano existence theorem, Exercise

Peano surface, Exercise

permutation, Paragraph

Picard iterate, Subsection

Picard iteration, Subsection

piecewise continuously differentiable path, Definition

piecewise smooth, Paragraph

piecewise smooth boundary, Definition

piecewise smooth path, Definition

piecewise smooth reparametrization, Definition

Poincaré lemma, Theorem

pointwise bounded, Definition

pointwise convergence, Definition Paragraph

complex series, Paragraph

pole, Paragraph

polynomial, Proposition

positive, Definition

positively oriented, Definition

potential, Paragraph

power series, Paragraph

power set, Definition

preserve orientation, Definition

principle of induction, Paragraph

principle of strong induction, Theorem

product rule, Proposition

proper, Exercise

proper subset, Item

pseudometric space, Item

quotient rule, Proposition

range, Definition

range of a sequence, Paragraph

ratio test for sequences, Lemma

ratio test for series, Proposition

rational function, Paragraph

rational functions, Paragraph

rational numbers, Item

real algebra, Definition

real axis, Paragraph

real numbers, Paragraph

real part, Definition

real vector space, Definition

real-analytic, Paragraph

rearrangement of a series, Paragraph

rectangle, Definition

refinement of a partition, Definition Definition

reflexive relation, Item

relation, Definition

relative maximum, Definition Paragraph

relative minimum, Definition Paragraph

remainder term in Taylor’s formula, Paragraph

removable discontinuity, Example

reparametrization, Definition

restriction, Definition

reverse orientation, Definition

reverse triangle inequality, Item

Riemann integrable, Definition Definition

complex-valued function, Paragraph

Riemann integrable on \(S\), Definition

Riemann integral, Definition Definition

Riemann--Lebesgue theorem, Theorem

Riemann–Lebesgue Lemma, Exercise

Rolle’s theorem, Theorem

root test, Paragraph

scalars, Paragraph

second derivative, Paragraph

second derivative test, Proposition

self-adjoint, Definition

separates points, Item

sequence, Definition Definition

sequentially compact, Paragraph

series, Definition

set, Definition

set building notation, Paragraph

set theory, Section

set-theoretic difference, Item

set-theoretic function, Paragraph

simple path, Definition

simply connected, Paragraph

sinc function, Example

sine, Paragraph

singularity, Paragraph

slope field, Paragraph

smooth path, Definition

smooth reparametrization, Definition

span, Definition

spectral radius, Paragraph

sphere, Example

squeeze lemma, Lemma

standard basis, Paragraph

standard metric on \(\R\), Example

standard metric on \(\R^n\), Example

star-shaped domain, Definition

step function, Paragraph

Stone--Weierstrass

complex version, Theorem

real version, Theorem

strict relative maximum, Paragraph

strict relative minimum, Paragraph

strictly decreasing, Definition Paragraph

strictly increasing, Definition Paragraph

strictly monotone function, Definition

subadditive, Exercise

subcover, Paragraph

subrectangle, Definition

subsequence, Definition Definition

subset, Item

subspace, Definition Definition

subspace metric, Paragraph

subspace topology, Paragraph

sup norm, Definition

support, Paragraph

supremum, Item

supremum norm, Paragraph

surjection, Definition

surjective, Definition

symmetric, Paragraph

symmetric difference, Exercise

symmetric group, Paragraph

symmetric partial sums, Paragraph

symmetric relation, Item

symmetry of a metric, Item

tail of a sequence, Definition

tail of a series, Paragraph

tangent vector, Paragraph

Taylor polynomial, Definition

Taylor series, Paragraph

Taylor’s theorem, Theorem

real-analytic, Theorem

Tonelli for sums, Exercise

topology, Paragraph

total derivative, Definition

totally bounded, Exercise

totally disconnected, Exercise

transformation, linear, Definition

transitive relation, Item

complex numbers, Paragraph

norms, Item

trichotomy, Item

trigonometric polynomial, Paragraph

type I domain, Paragraph

type II domain, Paragraph

type III domain, Paragraph

unbounded closed intervals, Paragraph

unbounded interval, Paragraph

unbounded open intervals, Paragraph

uncountable, Definition

uniform convergence, Definition Paragraph

uniform convergence on compact subsets, Exercise

uniform norm, Definition Paragraph

uniform norm convergence, Paragraph

uniformly bounded, Definition

uniformly Cauchy, Definition Paragraph

uniformly continuous, Definition

in a metric space, Definition

uniformly equicontinuous, Definition

union, Item

unit sphere, Example

universe, Paragraph

upper bound, Item

upper Darboux integral, Definition Definition

upper Darboux sum, Definition Definition

upper triangular matrix, Paragraph

vanishes at no point, Item

vector, Paragraph

vector field, Definition Example

vector space, Definition

vector subspace, Definition

Venn diagram, Paragraph

volume, Paragraph

volume of rectangles, Definition

vortex vector field, Example