-
Abel's convergence test, 247
-
Abel's theorem for power series, 249, 322
-
Absolute value
-
in an ordered field, 26
-
in En, 64
-
in Euclidean spaces, 88
-
in normed linear spaces, 90
-
Absolutely continuous functions (weakly), 309
-
Absolutely convergent series of functions, 237
-
rearrangement of, 238
-
tests for, 239
-
Accumulation points, 115. See also Cluster point
-
Additivity
-
of definite integrals, 282
-
of total variation, 301
-
of volume of intervals in En, 79
-
Alternating series, 248
-
Admissible change of variable, 165
-
Angle between vectors in En, 70
-
Antiderivative, 278. See also Integral, indefinite
-
Antidifferentiation, 278. See also Integration
-
Arcs, 211
-
as connected sets, 214
-
endpoints of, 211
-
length of, 301, 311
-
rectifiable, 309
-
simple, 211
-
Archimedean field, see Field, Archimedean
-
Archimedean property, 43
-
Arcwise connected set, 211
-
Arithmetic-geometric mean, Gauss's, 134
-
Associative laws
-
in a field, 23
-
of vector addition in En, 65
-
Axioms
-
of arithmetic in a field, 23
-
of a metric, 95
-
of order in an ordered field, 24
-
Back to Top
-
Basic unit vector in En, 64
-
Bernoulli inequalities, 33
-
Binary operations, 12. See also Functions
-
Binomial theorem, 34
-
Bolzano theorem, 205
-
Bolzano-Weierstrass theorem, 136
-
Boundary
-
of intervals in En, 77
-
of sets in metric spaces, 108
-
Bounded
-
functions on sets in metric spaces, 111
-
sequences in metric spaces, 111
-
sets in metric spaces, 109
-
sets in ordered fields, 36
-
variation, 303
-
left-bounded sets in ordered fields, 36
-
right-bounded sets in ordered fields, 36
-
totally bounded sets in a metric space, 188
-
uniformly bounded sequences of functions, 234
-
Back to Top
-
C (the complex field), 80
-
complex numbers, 81; see also Complex numbers
-
Cartesian coordinates in, 83
-
de Moivre's formula, 84
-
imaginary numbers in, 81
-
imaginary unit in, 81
-
is not an ordered field, 82
-
polar coordinates in, 83
-
real points in, 81
-
real unit in, 81
-
Cn (complex n-space), 87
-
as a Euclidean space, 88
-
as a normed linear space, 91
-
componentwise convergence of sequences in, 121
-
dot products in, 87
-
standard norm in, 91
-
Cantor's diagonal process, 21. See also Sets
-
Cantor's function, 186
-
Cantor's principle of nested closed sets, 188
-
Cantor's set, 120
-
Cartesian coordinates in C, 83
-
Cartesian product of sets, 2
-
intervals in En as Cartesian products of intervals in E1, 76
-
Cauchy criterion
-
for function limits, 162
-
for uniform convergence of sequences of functions, 231
-
Cauchy form of the remainder term of Taylor expansions, 291
-
Cauchy sequences in metric spaces, 141
-
Cauchy's convergence criterion for sequences in metric spaces, 143
-
Cauchy's laws of the mean, 261
-
Cauchy-Schwarz inequality
-
in En, 67
-
in Euclidean spaces, 88
-
Center of an interval in En, 77
-
Change of variable, admissible, 165
-
Chain rule for differentiation of composite functions, 255
-
Change of variables in definite integrals, 282
-
Characteristic functions of sets, 323
-
Clopen
-
sets in metric spaces, 103
-
Closed
-
curve, 211
-
globe in a metric space, 97
-
interval in an ordered field, 37
-
interval in En, 77
-
line segment in En, 72
-
sets in metric spaces, 103, 138
-
Closures of sets in metric spaces, 137
-
Closure laws
-
in a field, 23
-
in En, 65
-
of integers in a field, 35
-
of rationals in a field, 35
-
Cluster points
-
of sequences in E*, 60
-
of sequences and sets in metric spaces, 115
-
Commutative laws
-
in a field, 23
-
of addition of vectors in En, 65
-
of inner products of vectors in En, 67
-
Compact sets, 186, 193
-
Cantor's principle of nested closed sets, 188
-
are totally bounded, 188
-
in E1, 195
-
continuity on, 194
-
generalized Heine-Borel theorem, 193
-
Heine-Borel theorem, 324
-
sequentially, 186
-
Comparison test, 239
-
refined, 245
-
Complement of a set, 2
-
Complete
-
metric spaces, 143
-
ordered fields, 38; see also Field, complete ordered
-
Completeness axiom, 38
-
Completion of metric spaces, 146
-
Complex exponential, 173
-
derivatives of the, 256
-
Complex field, seeC
-
Complex functions, 170
-
Complex numbers, 81. See also C
-
conjugate of, 81
-
imaginary part of, 81
- nth roots of, 85
-
polar form of, 83
-
real part of, 81
-
trigonometric form of, 83
-
Complex vector spaces, 87
-
Componentwise
-
continuity of functions, 172
-
convergence of sequences, 121
-
differentiation, 256
-
integration, 282
-
limits of functions, 172
-
Composite functions, 163
-
chain rule for derivatives of, 255
-
continuity of, 163
-
Concurrent sequences, 144
-
Conditionally convergent series of functions, 237
-
rearrangement of, 250
-
Conjugate of complex numbers, 81
-
Connected sets, 212
-
arcs as, 214
-
arcwise-, 211
-
curves as, 214
-
polygon-, 204
-
Continuous functions
-
on metric spaces, 149
-
differentiable functions are, 252
-
left, 153
-
relatively, 152
-
right, 153
-
uniformly, 197
-
weakly absolutely continuous, 309
-
Continuity. See also Continuous functions
-
componentwise, 172
-
in one variable, 174
-
jointly, 174
-
of addition and multiplication in E1, 168
-
of composite functions, 163
-
of inverse functions, 195, 207
-
of the exponential function, 184
-
of the logarithmic function, 208
-
of the power function, 209
-
of the standard metric on E1, 168
-
of the sum, product, and quotient of functions, 170
-
on compact sets, 194
-
sequential criterion for, 161
-
uniform, 197
-
Contracting sequence of sets, 17
-
Contraction mapping, 198
-
Convergence of sequences of functions
-
Cauchy criterion for uniform, 231
-
convergence of integrals and derivatives, 315
-
pointwise, 228
-
uniform, 228
-
Convergence radius of power series, 243
-
Convergence tests for series
-
Abel's test, 247
-
comparison test, 239
-
Dirichlet test, 248
-
integral test, 327
-
Leibniz test for alternating series, 248
-
ratio test, 241
-
refined comparison test, 245
-
root test, 241
-
Weierstrass M-test for functions, 240
-
Convergent
-
absolutely convergent series of functions, 237
-
conditionally convergent series of functions, 237
-
sequences of functions, 228; see also Limits of sequences of functions
-
sequences in metric spaces, 115
-
series of functions, 228; see also Limits of series of functions
-
Convex sets, 204
-
piecewise, 204
-
Coordinate equations of a line in En, 72
-
Countable set, 18
-
rational numbers as a, 19
-
Countable union of sets, 20
-
Covering, open, 192
-
Cross product of sets, 2
-
Curves, 211
-
as connected sets, 214
-
closed, 211
-
length of, 300
-
parametric equations of, 212
-
tangent to, 257
-
Back to Top
-
Darboux property, 203
-
Bolzano theorem, 205
-
of the derivative, 265
-
de Moivre's formula, 84
-
Definite integrals, 279
-
additivity of, 282
-
change of variables in, 282
-
dominance law for, 284
-
first law of the mean for, 285
-
integration by parts, 281
-
linearity of, 280
-
monotonicity law for, 284
-
weighted law of the mean for, 286, 326
-
Degenerate intervals in En, 78
-
Degree
-
of a monomial, 173
-
of a polynomial, 173
-
Deleted globes about points in metric spaces, 150
-
Dense subsets in metric spaces, 139
-
Density
-
of an ordered field, 45
-
of rationals in an Archimedean field, 45
-
Dependent vectors
-
in En, 69
-
Derivatives of functions on E1, 251
-
convergence of, 315
-
Darboux property of, 265
-
derivative of the exponential function, 264
-
derivative of the inverse function, 263
-
derivative of the logarithmic function, 263
-
derivative of the power function, 264
-
with extended-real values, 259
-
left, 252
-
one-sided, 252
-
right, 252
-
Derived functions on E1, 251
- nth, 252
-
Diagonal of an interval in En, 77
-
Diagonal process, Cantor's, 21. See also Sets
-
Diameter
-
of sets in metric spaces, 109
-
Difference
-
of elements of a field, 26
-
of sets, 2
-
Differentials of functions on E1, 288
-
of order n, 289
-
Differentiable functions on E1, 251
-
Cauchy's laws of the mean, 261
-
cosine function, 337
-
are continuous, 252
-
exponential function, 333
-
infinitely, 292
-
logarithmic function, 332
-
n-times continuously, 292
-
n-times, 252
-
nowhere, 253
-
Rolle's theorem, 261
-
sine function, 337
-
Differentiation, 251
-
chain rule for, 255
-
componentwise, 256
-
of power series, 319
-
rules for sums, products, and quotients, 256
-
termwise differentiation of series, 318
-
Directed
-
lines in En, 74
-
planes in En, 74
-
Direction vectors
of lines in En, 71
-
Dirichlet function, 155, 329
-
Dirichlet test, 248
-
Disconnected sets, 212
-
totally, 217
-
Discontinuity points of functions on metric spaces, 149
-
Discontinuous functions on metric spaces, 149
-
Discrete
-
metric, 96
-
metric space, 96
-
Disjoint sets, 2
-
Distance
-
between a point and a plane in En, 76
-
between sets in metric spaces, 110
-
between two vectors in En, 64
-
between two vectors in Euclidean spaces, 89
-
in normed linear spaces, 92
-
norm-induced, 92
-
translation-invariant, 93
-
Distributive laws
-
in En, 65
-
in a field, 24
-
of inner products of vectors in En,
67
-
of union and intersection of sets, 7
-
Divergent
-
sequences in metric spaces, 115
-
Domain
-
of a relation, 9
-
of a sequence, 15
-
space of functions on metric spaces, 149
-
Double limits of functions, 219, 221
-
Double sequence, 20, 222, 223
-
Dot product
-
in Cn, 87
-
in En, 64
-
Duality laws, de Morgan's, 3. See also Sets
-
Back to Top
-
e (the number), 122, 165, 293
-
E1 (the real numbers), 23. See alsoField, complete ordered
-
associative laws in, 23
-
axioms of arithmetic in, 23
-
axioms of order in, 24
-
closure laws in, 23
-
commutative laws in, 23
-
continuity of addition and multiplication in, 168
-
continuity of the standard metric on, 168
-
distributive law in, 24
-
inverse elements in, 24
-
monotonicity in, 24
-
neighborhood of a point in, 58
-
natural numbers in, 28
-
neutral elements in, 23
-
transitivity in, 24
-
trichotomy in, 24
-
En (Euclidean n-space), 63. See also Vectors in En
-
convex sets in, 204
-
as a Euclidean space, 88
-
as a normed linear space, 91
-
associativity of vector addition in, 65
-
additive inverses of vector addition, 65
-
basic unit vector in, 64
-
Bolzano-Weierstrass theorem, 136
-
Cauchy-Schwarz inequality in, 67
-
closure laws in, 65
-
commutativity of vector addition in, 65
-
componentwise convergence of sequences in, 121
-
distributive laws in, 65
-
globe in, 76
-
hyperplanes in, 72; see also Planes in En
-
intervals in, 76; see also Intervals in En
-
line segments in, 72; see also Line segments in En
-
linear functionals on, 74, 75; see also Linear functionals on En
-
lines in, 71; see also Lines in En
-
neutral element of vector addition in, 65
-
planes in, 72; see also Planes in En
-
point in, 63
-
scalar of, 64
-
scalar product in, 64
-
sphere in, 76
-
standard metric in, 96
-
standard norm in, 91
-
triangle inequality of the absolute value in, 67
-
triangle inequality of the distance in, 68
-
unit vector in, 65
-
vectors in, 63
-
zero vector in, 63
-
E* (extended real numbers), 53
-
as a metric space, 98
-
cluster point of a sequence in, 60
-
globes in, 98
-
indeterminate expressions in, 178
-
intervals in, 54
-
limits of sequences in, 58
-
metrics for, 99
-
neighborhood of a point in, 58
-
operations in, 177
-
unorthodox operations in, 180
-
Edge-lengths of an interval in En, 77
-
Elements of a set, 1
-
Empty set, 1
-
Endpoints
-
of an interval in En, 77
-
of line segments in En, 72
-
Equality of sets, 1
-
Equicontinuous functions, 236
-
Equivalence class relative to an equivalence relation, 13
-
generator of an, 13
-
representative of an, 13
-
Equivalence relation, 12
-
equivalence class relative to an, 13
-
Euclidean n-space, see En
-
Euclidean spaces, 88
-
as normed linear spaces, 91
-
absolute value in, 88
-
Cn as a, 88
-
Cauchy-Schwarz inequality in, 88
-
distance in, 89
-
En as a, 88
-
line segments in, 89
-
lines in, 89
-
planes in, 89
-
triangle inequality in, 88
-
Exact primitive, 278
-
Existential quantifier, 4
-
Expanding sequence of sets, 17
-
Exponential, complex, 173
-
Exponential function, 183, 333
-
continuity of the, 184
-
derivative of the, 264
-
inverse of the, 208
-
Extended real numbers, see E*
-
Back to Top
-
Factorials, definition of, 31
-
Family of sets, 3
-
intersection of a, 3
-
union of a, 3
-
Fields, 25
-
associative laws in, 23
-
axioms of arithmetic in, 23
-
binomial theorem, 34
-
closure laws in, 23
-
commutative laws in, 23
-
difference of elements in, 26
-
distributive law in, 24
-
first induction law in, 28
-
inductive definitions in, 31
-
inductive sets in, 28
-
integers in, 34
-
inverse elements in, 24
-
irrationals in, 34
-
Lagrange identity in, 71
-
natural elements in, 28
-
neutral elements in, 23
-
quotients of elements in, 26
-
rational subfields of, 35
-
rationals in, 34
-
Fields, Archimedean, 43. See also Fields, ordered
-
density of rationals in, 45
-
integral parts of elements of, 44
-
Fields, complete ordered, 38. See also Field, Archimedean
-
Archimedean property of, 43
-
completeness axiom, 38
-
density of irrationals in, 51
-
existence of irrationals in, 46
-
powers with rational exponents in, 47
-
powers with real exponents in, 50
-
principle of nested intervals in, 42
-
roots in, 46
-
Fields, ordered, 25. See also Field
-
absolute value in, 26
-
axioms of order in, 24
-
Bernoulli inequalities in, 33
-
bounded sets in, 36
-
closed intervals in, 37
-
density of, 45
-
greatest lower bound (glb) of sets in, 38
-
half-closed intervals in, 37
-
half-open intervals in, 37
-
infimum (inf) of sets in, 38
-
intervals in, 37
-
least upper bound (lub) of sets in, 37
-
monotonicity in, 24
-
negative elements in, 25
-
open intervals in, 37
-
positive elements in, 25
-
rational subfield in, 35
-
second induction law in, 30
-
supremum (sup) of sets in, 38
-
transitivity in, 24
-
trichotomy in, 24
-
well-ordering of naturals in, 30
-
Finite
-
increments law, 271
-
intervals, 54
-
sequence, 16
-
set, 18
-
First
-
induction law, 28
-
law of the mean, 285
-
Functions, 10. See also Functions on E1 and Functions on metric spaces
-
binary operations, 12
-
bounded, 96
-
Cantor's function, 186
-
characteristic, 323
-
complex, 170
-
Dirichlet function, 155, 329
-
equicontinuous, 236
-
graphs of, 153
-
isometry, 201
-
limits of sequences of, see Limits of sequences of functions
-
limits of series of, see Limits of series of functions
-
monotone, 181
-
nondecreasing, 181
-
nonincreasing, 181
-
one-to-one, 10
-
onto, 11
-
product of, 170
-
quotient of, 170
-
real, 170
-
scalar-valued, 170
-
sequences of, 227; see also Sequences of functions
-
series of, 228; see also Limits of series of functions
-
signum function (sgn), 156
-
strictly monotone, 182
-
sum of, 170
-
function value, 10
-
uniformly continuous, 197
-
vector-valued, 170
-
Functions on E1
-
antiderivatives of, 278
-
definite integrals of, 279
-
derivatives of, 251
-
derived, 251
-
differentials of, 288; see also Differentials of functions on E1
-
differentiable, 251; see also Differentiable functions on E1
-
exact primitives of, 278
-
of bounded variation, 303
-
indefinite integrals of, 278
-
integrable, 278; see also Integrable functions on E1
-
length of, 301
-
Lipschitz condition for, 258
-
negative variation functions for, 308
-
nowhere differentiable, 253
-
positive variation functions for, 308
-
primitives of, 278
-
regulated, see Regulated functions
-
simple step, 323
-
step, 323
-
total variation of, 301
-
weakly absolutely continuous, 309
-
Functions on metric spaces,149
-
bounded, 111
-
continuity of composite, 163
-
continuity of the sum, product, and quotient of, 170
-
continuous, 149
-
discontinuous, 149
-
discontinuity points of, 149
-
domain space of, 149
-
limits of, 150
-
projection maps, 174, 198, 226
-
range space of, 149
-
Back to Top
-
General term of a sequence, 16
-
Generator of an equivalence class, 13
-
Geometric series
-
limit of, 128, 236
-
sum of n terms of a, 33
-
Globes
-
closed globes in metric spaces, 97
-
deleted globes about points in metric spaces, 150
-
in En, 76
-
in E*, 98
-
open globes in metric space, 97
-
Graphs of functions, 153
-
Greatest lower bound (glb) of a set in an ordered field, 38
-
Back to Top
-
Half-closed
-
interval in an ordered field, 37
-
interval in En, 77
-
line segment in En, 72
-
Half-open
-
interval in an ordered field, 37
-
interval in En, 77
-
line segment in En, 72
-
Harmonic series, 241
-
Hausdorff property, 102
-
Heine-Borel theorem, 324
-
generalized, 193
-
Holder's inequality, 93
-
Hyperharmonic series, 245, 329
-
Hyperplanes in En, 72. See also Planes in En
-
Back to Top
-
iff (``if and only if''), 1
-
Image
-
of a set under a relation, 9
-
Imaginary
-
part of complex numbers, 81
-
numbers in C, 81
-
unit in C, 81
-
Inclusion relation of sets, 1
-
Increments
-
finite increments law, 271
-
of a function, 254
-
Independent
-
vectors in En, 70
-
Indeterminate expressions in E*, 178
-
Index notation, 16. See also Sequence
-
Induction, 27
-
first induction law, 28
-
inductive definitions, 31; see also Inductive definitions
-
proof by, 29
-
second induction law, 30
-
Inductive definitions, 31
-
factorial, 31
-
powers with natural exponents, 31
-
ordered n-tuple, 32
-
products of n field elements, 32
-
sum of n field elements, 32
-
Inductive sets in a field, 28
-
Infimum (inf) of a set in an ordered field, 38
-
Infinite
-
countably, 21
-
intervals, 54
-
sequence, 15
-
set, 18
-
Infinity
-
plus and minus, 53
-
unsigned, 179
-
Inner products
of vectors in En, 64
-
commutativity of, 67
-
distributive law of, 67
-
Integers in a field, 34
-
closure of addition and multiplication of, 35
-
Integrability, sufficient conditions for, 322. See also Regulated functions on intervals in E1
-
Integrable functions on E1, 278. See also Regulated functions on intervals in E1
-
Dirichlet function, 329
-
primitively, 278
-
Integral part of elements of Archimedean fields, 44
-
Integral test of convergence of series, 315
-
Integrals
-
convergence of, 315
-
definite, 279; see also Definite integrals
-
indefinite, 278
-
Integration, 278
-
componentwise, 282
-
by parts, 281
-
of power series, 319
-
Interior
-
of a set in a metric space, 101
-
points of a set in a metric space, 101
-
Intermediate value property, 203
-
Intersection
-
of a family of sets, 3
-
of closed sets in metric spaces, 104
-
of open sets in metric spaces, 103
-
of sets, 2
-
Intervals in En, 76
-
boundary of, 77
-
center of, 77
-
closed, 77
-
degenerate, 78
-
diagonal of, 77
-
edge-lengths of, 77
-
endpoints of, 77
-
half-closed, 77
-
half-open, 77
-
midpoints of, 77
-
open, 77
-
principle of nested, 189
-
volume of, 77
-
Intervals in E1
-
partitions of, 300
-
Intervals in E*, 54
-
finite, 54
-
infinite, 54
|
-
Intervals in an ordered field, 37
-
closed, 37
-
half-closed, 37
-
half-open, 37
-
open, 37
-
principle of nested, 42
-
Inverse elements
-
in a field, 24
-
of vector addition in En, 64, 65
-
Inverse function, see Inverse of a relation
-
continuity of the, 195, 207
-
derivative of the, 263
-
Inverse
image of a set under a relation, 9
-
Inverse pair, 8
-
Inverse of a relation, 8
-
Irrationals
-
density of irrationals in a complete field, 51
-
existence of irrationals in a complete field, 46
-
in a field, 34
-
Isometric metric spaces, 146
-
Isometry, 201. See also Functions
-
Iterated limits of functions, 221, 221
-
Back to Top
-
Jumps of regulated functions, 330
-
Back to Top
-
Kuratowski's definition of ordered pairs, 7
-
Back to Top
-
Lagrange form of the remainder term of Taylor expansions, 291
-
Lagrange identity, 71
-
Lagrange's law of the mean, 262
-
Laws of the mean
-
Cauchy's, 261
-
first, 285
-
Lagrange's, 262
-
second, 286, 326
-
weighted, 286, 326
-
Leading term of a polynomial, 173
-
Least upper bound (lub) of a set in an ordered field, 37
-
Lebesgue number of a covering, 192
-
Left
-
bounded sets in an ordered field, 36
-
continuous functions, 153
-
derivatives of functions, 252
-
jump of a function, 184
-
limits of functions, 153
-
Leibniz
-
formula for derivatives of a product, 256
-
test for convergence of alternating series, 248
-
Length
-
function, 308
-
of arcs, 301, 311
-
of curves, 300
-
of functions, 301
-
of line segments in En, 72
-
of polygons, 300
-
of vectors in En, 64
-
L'Hospital's rule, 266
-
Limits of functions
-
Cauchy criterion for, 162
-
componentwise, 172
-
double, 219, 221
-
iterated, 221, 221
-
jointly, 174
-
left, 153
-
on E*, 151
-
in metric spaces, 150
-
limits in one variable, 174
-
L'Hospital's rule, 266
-
relative, 152
-
relative, over a line, 174
-
right, 153
-
subuniform, 225
-
uniform, 220, 230
-
Limits of sequences
-
in E1, 5, 54
-
in E*, 55, 58, 152
-
in metric spaces, 115
-
lower, 56
-
subsequential limits, 135
-
upper, 56
-
Limits of sequences of functions
-
pointwise, 228
-
uniform, 228
-
Limits of series of functions
-
pointwise, 228
-
uniform, 228
-
Weierstrass M-test, 240
-
Linear combinations of vectors in En, 66
-
Line segments in En, 72
-
closed, 72
-
endpoints of, 72
-
half-closed, 72
-
half-open, 72
-
length of, 72
-
midpoint of, 72
-
open, 72
-
principle of nested, 205
-
Linear functionals on En, 74, 75
-
equivalence between planes and nonzero, 76
-
representation theorem for, 75
-
Linear polynomials, 173
-
Linear spaces, see Vector spaces
-
Linearity of the definite integral, 280
-
Lines in En, 71
-
coordinate equations of, 72
-
directed, 74
-
direction vectors of, 71
-
normalized equation of, 73
-
parallel, 74
-
parametric equations of, 72
-
perpendicular, 74
-
symmetric form of the normal equations of, 74
-
Lipschitz condition, 258
-
Local
-
maximum and minimum of functions, 260
-
Logarithmic function, 208
-
continuity of the, 208
-
derivative of the, 263
-
integral definition of the, 331
-
as the inverse of the exponential function, 208
-
natural logarithm, 208
-
properties of the, 332
-
Logical formula, negation of a, 5
-
Logical quantifier, see Quantifier, logical
-
Lower bound of a set in an ordered field, 36
-
Lower limit of a sequence, 56
-
Back to Top
-
Maclaurin series, 294
-
Mapping, see Function
-
contraction, 198
-
projection, 174, 198, 226
-
Master set, 2
-
Maximum
-
local, of a function, 260, 294
-
of a set in an ordered field, 36
-
Mean, laws of. See Laws of the mean
-
Metrics, 95. See also Metric spaces
-
axioms of, 95
-
discrete, 96
-
equivalent, 219
-
for E*, 99
-
standard metric in En, 96
-
Metric spaces, 95. See also Metrics
-
accumulation points of sets or sequences in, 115
-
boundaries of sets in, 108
-
bounded functions on sets in, 111
-
bounded sequences in, 111
-
bounded sets in, 109
-
Cauchy sequences in, 141
-
Cauchy's convergence criterion for sequences in, 143
-
clopen sets in, 103
-
closed balls in, 97
-
closed sets in, 103, 138
-
closures of sets in, 137
-
compact sets in, 186
-
complete, 143
-
completion of, 146
-
concurrent sequences in, 144
-
connected, 212
-
constant sequences in, 116
-
continuity of the metric on, 223
-
convergent sequences in, 115
-
cluster points of sets or sequences in, 115
-
deleted globes about points in, 150
-
diameter of sets in, 109
-
disconnected, 212
-
dense subsets in, 139
-
discrete, 96
-
distance between sets in, 110
-
divergent sequences in, 115
-
En as a metric space, 96
-
E* as a metric space, 98
-
functions on, 149; see also Functions on metric spaces
-
Hausdorff property in, 102
-
interior of a set in a, 101
-
interior points of sets in, 101
-
isometric, 146
-
limits of sequences in, 115
-
nowhere dense sets in, 141
-
open balls in, 97
-
open sets in, 101
-
open globes in, 97
-
neighborhoods of points in, 101
-
perfect sets in, 118
-
product of, 218
-
sequentially compact sets in, 186
-
spheres in, 97
-
totally bounded sets in, 113
-
Midpoints
-
of line segments in En, 72
-
of intervals in En, 77
-
Minimum
-
local, of a function, 260, 294
-
of a set in an ordered field, 36
-
Minkowski inequality, 94
-
Monomials in n variables, 173. See also Polynomials in n variables
-
degree of, 173
-
Monotone sequence of numbers, 17
-
nondecreasing, 17
-
nonincreasing, 17
-
strictly, 17
-
Monotone functions, 181
-
left and right limits of, 182
-
nondecreasing, 181
-
nonincreasing, 181
-
strictly, 182
-
Monotone sequence of sets, 17
-
Monotonicity
-
in an ordered field, 24
-
of definite integrals, 284
-
Moore-Smith theorem, 223
-
de Morgan's duality laws, 3. See also Sets
-
Back to Top
-
Natural elements in a field, 28
-
well-ordering of naturals in an ordered field, 30
-
Natural numbers in E1, 28
-
Negation of a logical formula, 5
-
Negative
-
elements of an ordered field, 25
-
variation functions, 308
-
Neighborhood
-
of a point in E1, 58
-
of a point in E*, 58
-
of a point in a metric space, 101
-
Neutral elements
-
in a field, 23
-
of vector addition in En, 65
-
Nondecreasing
-
functions, 181
-
sequences of numbers, 17
-
Nonincreasing
-
functions, 181
-
sequences of numbers, 17
-
Normal to a plane in En, 73
-
Normalized equations
-
of a line, 73
-
of a plane, 73
-
Normed linear spaces, 90
-
absolute value in, 90
-
Cn as a, 91
-
distances in, 92
-
En as a, 91
-
Euclidean spaces as, 91
-
norm in, 90
-
translation-invariant distances in, 93
-
triangle inequality in, 90
-
Norms
-
in normed linear spaces, 90
-
standard norm in Cn, 91
-
standard norm in En, 91
-
Nowhere dense sets in metric spaces, 141
-
Back to Top
-
Open
-
ball in a metric space, 97
-
covering, 192
-
globe in a metric space, 97
-
interval in an ordered field, 37
-
interval in En, 77
-
line segment in En, 72
-
sets in a metric space, 101
-
Ordered field, see Field, ordered
-
Ordered n-tuple, 1
-
inductive definition of an, 32
-
Ordered pair, 1
-
inverse, 8
-
Kuratowski's definition of an, 7
-
Orthogonal vectors in En, 65
-
Orthogonal projection
-
of a point onto a plane in En, 76
-
Osgood's theorem, 221, 223
-
Back to Top
-
Parallel
-
lines in En, 74
-
planes in En, 74
-
vectors in En, 65
-
Parametric equations
-
of curves in En, 212
-
of lines in En, 72
-
Partitions of intervals in E1, 300
-
refinements of, 300
-
Pascal's law, 34
-
Peano form of the remainder term of Taylor expansions, 296
-
Perfect sets in metric spaces, 118
-
Cantor's set, 120
-
Perpendicular
-
lines in En, 74
-
planes in En, 74
-
vectors in En, 65
-
Piecewise convex sets, 204
-
Planes in En, 72
-
directed, 74
-
distance between points and, 76
-
equation of, 73
-
equivalence of nonzero linear functionals and, 76
-
general equation of, 73
-
normal to, 73
-
normalized equations of, 73
-
orthogonal projection of a point onto, 76
-
parallel, 74
-
perpendicular, 74
-
Point in En, 63
-
distance from a plane to a, 76
-
orthogonal projection onto a plane, 76
-
Pointwise limits
-
of sequences of functions, 228
-
of series of functions, 228
-
Polar coordinates in C, 83
-
Polar form of complex numbers, 83
-
Polygons
-
connected sets, 204
-
joining two points, 204
-
length of, 300
-
Polygon-connected sets, 204
-
Polynomials in n variables, 173
-
continuity of, 173
-
degree of, 173
-
leading term of, 173
-
linear, 173
-
Positive
-
elements of an ordered field, 25
-
variation functions, 308
-
Power function, 208
-
continuity of the, 209
-
derivative of the, 264
-
Power series, 243
-
Abel's theorem for, 249
-
differentiation of, 319
-
integration of, 319
-
radius of convergence of, 243
-
Taylor series, 292
-
Powers
-
with natural exponents in a field, 31
-
with rational exponents in a complete field, 47
-
with real exponents in a complete field, 50
-
Primitive, 278. See also Integral, indefinite
-
exact, 278
-
Principle of nested
-
closed sets, 188
-
intervals in complete ordered fields, 189
-
intervals in En, 189
-
intervals in ordered fields, 42
-
line segments, 205
-
Products of functions, 170
-
derivatives of, 256
-
Leibniz formula for derivatives of, 256
-
Product of metric spaces, 218
-
Projection maps, 174, 198, 226
-
Proper subset of a set, 1
-
Back to Top
-
Quantifier, logical, 3
-
existential, 4
-
universal, 4
-
Quotient of elements of a field, 26
-
Quotient of functions, 170
-
derivatives of, 256
-
Back to Top
-
Radius of convergence of a power series, 243
-
Range
-
of a relation, 9
-
of a sequence, 16
-
space of functions on metric spaces, 149
-
Ratio test for convergence of series, 241
-
Rational functions, 173
-
continuity of, 173
-
Rational numbers, 19
-
as a countable set, 19
-
Rationals
-
closure laws of, 35
-
density of rationals in an Archimedean field, 45
-
incompleteness of, 47
-
in a field, 34
-
as a subfield, 35
-
Real
-
functions, 170
-
numbers, see E1
-
part of complex numbers, 81
-
points in C, 81
-
vector spaces, 87
-
unit in C, 81
-
Rearrangement
-
of absolutely convergent series of functions, 238
-
of conditionally convergent series of functions, 250
-
Rectifiable
-
arc, 309
-
set, 303
-
Recursive definition, 31. See also
Inductive definition
-
Refined comparison test for convergence of series, 245
-
Refinements of partitions in E1, 300
-
Reflexive relation, 12
-
Regulated functions on intervals in E1, 323
-
approximation by simple step functions, 324
-
characteristic functions of intervals, 323
-
jumps of, 330
-
are integrable, 325
-
simple step functions, 323
-
Relation, 8. See also Sets
-
domain of a, 9
-
equivalence, 12
-
image of a set under a, 9
-
inverse, 8
-
inverse image of a set under a, 9
-
range of a, 9
-
reflexive, 12
-
symmetric, 12
-
transitive, 12
-
Relative
-
continuity of functions, 152, 174
-
limits of functions, 152, 174
-
Remainder term of Taylor expansions, 289
-
Cauchy form of the, 291
-
integral form of the, 289
-
Lagrange form of the, 291
-
Peano form of the, 296
-
Schloemilch-Roche form of the, 296
-
Representative of an equivalence class, 13
-
Right
-
bounded sets in an ordered field, 36
-
continuous functions, 153
-
derivatives of functions, 252
-
jump of a function, 184
-
limits of functions, 153
-
Rolle's theorem, 261
-
Root test for convergence of series, 241
-
Roots
-
in C, 85
-
in a complete field, 46
-
Back to Top
-
Scalar field of a vector space, 86
-
Scalar products
-
in En, 64
-
Scalar-valued functions, 170
-
Scalars
-
of En, 64
-
of a vector space, 86
-
Schloemilch-Roche form of the remainder term of Taylor expansions, 296
-
Second induction law, 30
-
Second law of the mean, 286, 326
-
Sequences, 15
-
bounded, 111
-
Cauchy, 141
-
Cauchy's convergence criterion for, 143
-
concurrent, 144
-
constant, 116
-
convergent, 115
-
divergent, 115
-
domain of, 15
-
double, 20, 222, 223
-
cluster points of sequences in E*, 60
-
finite, 16
-
general terms of, 16
-
index notation, 16
-
infinite, 15
-
limits of sequences in E1, 5, 54
-
limits of sequences in E*, 55, 58, 152
-
limits of sequences in metric spaces, 115
-
lower limits of, 56
-
monotone sequences of numbers, 17
-
monotone sequences of sets, 17
-
nondecreasing sequences of numbers, 17
-
nonincreasing sequences of numbers, 17
-
range of, 16
-
of functions, 227; see also Sequences of functions
-
strictly monotone sequences of numbers, 17
-
subsequences of, 17
-
subsequential limits of, 135
-
totally bounded, 188
-
upper limits of, 56
-
Sequences of functions
-
limits of, see Limits of sequences of functions
-
uniformly bounded, 234
-
Sequential criterion
-
for continuity, 161
-
for uniform continuity, 203
-
Sequentially compact sets, 186
-
Series. See also Series of functions
-
Abel's test for convergence of, 247
-
alternating, 248
-
geometric, 128, 236
-
harmonic, 241
-
hyperharmonic, 245, 329
-
integral test of convergence of, 327
-
Leibniz test for convergence of alternating series, 248
-
ratio test for convergence of, 241
-
refined comparison test, 245
-
root test for convergence of, 241
-
summation by parts, 247
-
Series of functions, 228; see also Limits of series of functions
-
absolutely convergent, 237
-
conditionally convergent, 237
-
convergent, 228
-
Dirichlet test, 248
-
differentiation of, 318
-
divergent, 229
-
integration of, 318
-
limit of geometric series, 128
-
power series, 243; see also Power series
-
rearrangement of, 238
-
sum of n terms of a geometric series, 33
-
Sets, 1
-
Cantor's diagonal process, 21
-
Cantor's set, 120
-
Cartesian product of, 2
-
characteristic functions of, 323
-
compact, 186, 193
-
complement of a set, 2
-
connected, 212
-
convex, 204
-
countable, 18
-
countable union of, 20
-
cross product of, 2
-
diagonal process, Cantor's, 21
-
difference of, 2
-
disjoint, 2
-
distributive laws of, 7
-
contracting sequence of, 17
-
elements of, 1
-
empty set, 1
-
equality of, 1
-
expanding sequence of, 17
-
family of, 3
-
finite, 18
-
inclusion relation of, 1
-
infinite, 18
-
intersection of a family of, 3
-
intersection of, 2
-
master set, 2
-
monotone sequence of, 17
-
de Morgan's duality laws, 3
-
perfect sets in metric spaces, 118
-
piecewise convex, 204
-
polygon-connected, 204
-
proper subset of a set, 1
-
rectifiable, 303
-
relation, 8
-
sequentially compact, 186
-
subset of a set, 1
-
superset of a set), 1
-
uncountable, 18
-
union of a family of, 3
-
union of, 2
-
Signum function (sgn), 156
-
Simple arcs, 211
-
endpoints of, 211
-
Simple step functions, 323
-
approximating regulated functions, 324
-
Singleton, 103
-
Span of a set of vectors in a vector space, 90
-
Sphere
-
in En, 76
-
in a metric space, 97
-
Step functions, 323
-
simple, 323
-
Strictly monotone functions, 182
-
Subsequence of a sequence, 17
-
Subsequential limits, 135
-
Subset of a set, 1
-
proper, 1
-
Subuniform limits of functions, 225
-
Sum of functions, 170
-
Summation by parts, 247
-
Superset of a set, 1
-
Supremum (sup) of a bounded set in an ordered field, 38
-
Symmetric relation, 12
-
Back to Top
-
Tangent
-
lines to curves, 257
-
vectors to curves, 257
-
unit tangent vectors, 314
-
Taylor. See also Taylor expansions
-
expansions, 289
-
polynomial, 289
-
series, 292; see also power series
-
series about zero (Maclaurin series), 294
-
Taylor expansions, 289. See also Remainder term of Taylor expansions
-
for the cosine function, 297
-
for the exponential function, 293
-
for the logarithmic function, 298
-
for the power function, 298
-
for the sine function, 297
-
Termwise
-
differentiation of series of functions, 318
-
integration of series of functions, 318
-
Total variation, 301
-
additivity of, 301
-
function, 308
-
Totally bounded sets in metric spaces, 113
-
Totally disconnected sets, 217
-
Transitive relation, 12
-
Transitivity in an ordered field, 24
-
Triangle inequality
-
in Euclidean spaces, 88
-
in normed linear spaces, 90
-
of the absolute value in En, 67
-
of the distance in En, 68
-
Trichotomy in an ordered field, 24
-
Trigonometric form of complex numbers, 83
-
Trigonometric functions
-
arcsine, 334
-
cosine, 336
-
integral definitions of, 334
-
sine, 336
-
Back to Top
-
Uncountable set, 18
-
Cantor's diagonal process, 21
-
the real numbers as a, 20
-
Uniform continuity, 197
-
sequential criterion for, 203
-
Uniform limits
-
of functions, 220, 230
-
of sequences of functions, 228
-
of series of functions, 228
-
Uniformly continuous functions, 197
-
Union
-
countable, 20
-
of a family of sets, 3
-
of closed sets in metric spaces, 104
-
of open sets in metric spaces, 103
-
of sets, 2
-
Unit vector
-
tangent, 314
-
in En, 65
-
Universal quantifier, 4
-
Unorthodox operations in E*, 180
-
Upper bound of a set in an ordered field, 36
-
Upper limit of a sequence, 56
-
Back to Top
-
Variation
-
bounded, 303
-
negative variation functions, 308
-
positive variation functions, 308
-
total; see Total variation
-
Vector-valued functions, 170
-
Vectors in En, 63
-
absolute value of, 64
-
angle between, 70
-
basic unit, 64
-
components of, 63
-
coordinates of, 63
-
dependent, 69
-
difference of, 64
-
distance between two, 64
-
dot product of two, 64
-
independent, 70
-
inner product of two, 64; see also
Inner products of vectors in En
-
inverse of, 65
-
length of, 64
-
linear combination of, 66
-
orthogonal, 65
-
parallel, 65
-
perpendicular, 65
-
sum of, 64
-
unit, 65
-
zero, 63
-
Vector spaces, 86
-
complex, 87
-
Euclidean spaces, 88
-
normed linear spaces, 90
-
real, 87
-
scalar field of, 86
-
span of a set of vectors in, 90
-
Volume of an interval in En, 77
-
additivity of the, 79
-
Back to Top
-
Weierstrass M-test for convergence of series, 240
-
Weighted law of the mean, 286, 326
-
Well-ordering property, 30
-
Back to Top
-
Zero vector in En, 63
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