References

[1] D. Bressoud. A Radical Approach to Real Analysis. Mathematical Association of America, Washington, D.C., 1994.

[2] P. Halmos. Naive Set Theory. D. Van Nostrand, Princeton, NJ, 1960.

[3] G. H. Hardy. A Course of Pure Mathematics. Cambridge University Press, Cambridge, UK, 9th edition, 1949.

[4] G.H. Hardy and E.M. Wright. An Introduction to the Theory of Num-bers. Oxford University Press, New York, 4th edition, 1960.

[5] LN. Herstein. Topics in Algebra. Wiley, New York, 2nd edition, 1975.

[6] J.L. Kelley. General Topology. Van Nostrand, Princeton, NJ, 1955.

[7] A.Ya. Khinchin. Continued Fractions. University of Chicago Press, Chicago, 3rd edition, 1964.

[8] T.W. Korner. Fourier Analysis. Cambridge University Press, Cam­bridge, UK, 1988.

[9] J.W. Milnor. Topology from the Differentiable Viewpoint. University Press of Virginia, Charlottesville, 1965.

[10J J.R. Munkres. Analysis on Manifolds. Addison-Wesley, Redwood City, CA, 1991.

[11] W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, New York, 3rd edition, 1976.

324 References

[12] G.E. Shilov and B.L. Gurevich. Integral, Measure and Derivative: A Unified Approach. Prentice-Hall, Englewood Cliffs, NJ, 1966.

[13] M. Spivak. Calculus on Manifolds. Addison-Wesley, Redwood City, CA,1965.

[14] O. Toeplitz. Calculus: A Genetic Approach. University of Chicago Press, Chicago, 1963.

[15] E.T. Whittaker and G.N. Watson. A Course of Modern Analysis. Cambridge University Press, Cambridge, 4th edition, 1962.

Index

a.a., see almost all Abel, N., 53, 73 absolutely convergent

integral, 116 series, 41

additive countably, 206 finitely, 203

adjoint map, 280 a.e., see almost everywhere Agnesi, M., 96 Alexandroff, P., 153 algebra

generated by, 205 of sets, 202

almost all, 221, 226 almost everywhere, 226 alternating tensor, 272 alternating tensors

basis for, 276 dimension of, 276 in R 3 , 279

alternation operator, 275 Ampere, A.M., 321 antiderivative, 110

antisymmetry, 273 Arbogast, L., 95 Archimedean property, 15 Archimedes, 26, 95 Arzela, C., 154 Arzela-Ascoli theorem, 145, 146 Ascoli, G., 154 axiom of choice, 217, 218

B n ,257 Baire, R., 153 Baire theorem, 136 ball, 128 Barrow, 1., 122 basis, 176

natural, 176 Bernoulli, Jakob, 53, 121 Bernoulli, Johann, 96 Bernstein, S., 174 bijective, 3 Bolzano, B., 53, 73, 153 Bolzano-Weierstrass property, 144 Bolzano-Weierstrass theorem, 35 Bonnet, 0., 96 Borel, E., 202, 221

326 Index

Borel algebra, 205 measure, 206 set, 205, 214, 215, 220, 224

Borel measure translation invariant, unique­

ness of, 239 Borel-Cantelli lemmma, 222 bound

lower, 14 upper, 14

boundary, 126 of manifold, 256

bounded, 142 totally, 142

Brouwer, L.E.J., 321 Bunyakovsky, V., 109

en, Coo, 88 C r ,254 Cantor, G., 26, 27,153, 174 Cantor function, 248 Cantor set, 149, 201, 248

dimension of, 211 measure of, 213

Caratheodory, C., 210, 221 cardinality, 6 Cartan, E., 284 Cartesian product, 2, 133 Cauchy, A., 72, 96, 121, 122 Cauchy condensation test, 43 Cauchy sequence, 36, 237 chain rule, 82, 184

classical notation, 187 change of variables

linear, 240 smooth, 112, 241

class C r , 190 closed,124 closed ball, 257 closure, 125 column vector, 177 compact, 140, 215

relatively, 144 sequentially, 144

compact set continuous image of, 141

complement, 2 relative, 1

complete, 236 metric space, 136 ordered field, 15

component, 148 condensation point, 150 conjugate space, 269 connected, 147, 186 connected component, 148 connected set

continuous image, 148 content, 121 continued fraction, 36-39 continuity from below, 206 continuous function, 57 continuous map, 126 contraction map, 137, 193 convergence

almost everywhere, 249 in measure, 249 mean square, 166 monotone, 225 pointwise, 62, 226, 229 uniform, 63, 226

convex coordinate patch, 293 convex function, 71, 78 convex set, 185, 250 convolution, 251 coordinate patch, 255 coset, 218 cosets, 277 countable, 6-8 countable additivity, 219 countable base, 131 countable subadditivity, 219 countably additive, 206 countably infinite, 6 countably subadditive, 209 covector, 269, 283 Cramer's rule, 181 critical point, 251 critical value, 251

cross product, 289 cube, 242 curl, 289 curve, 253 cut-off function, 299

.6. (Laplace operator), 314 155, see Kronecker 15 t5j, see Kronecker 15 Daniell, P.J., 252 Darboux, G., 84, 96 decreasing

function, 56 Dedekind, R., 21, 26 deformation, 262 DeMorgan's laws, 2, 212 dense, 16, 126, 218, 220 de Rham cohomology, 291 derivative, 74, 182

directional, 197 higher order, 88 of convex function, 80, 81 of exponential, 77 of polynomial, 76 of trigonometric function, 77

determinant, 181, 275 Jacobian, 182

diagonal matrix, 240 diameter, 142 diffeomorphism, 254 differentiable, 74, 182, 254 differential, 76, 182, 261

of form, see exterior differen­tial

differential calculus, 81 differential form, 285

closed, 291 exact, 291

differentiation of series, 114

Dini, V., 73 Dini's theorem, 64 Dirichlet, P.G.L., 53, 73, 174 Dirichlet principle, 317 Dirichlet test

Index 327

for integrals, 117 for series, 44

Dirichlet's theorem, 16, 218 disconnected, 147

totally, 149 distribution function, 248, 251 divergence, 288 divergence theorem, 312 domain, 3 dominated convergence theorem,

230 dual basis, 270, 282 dual space, 269 dummy variable, 245 Dynkin's 7r-A theorem, 238

€-net, 143 c(a), see sign of permutation c5, see Kronecker epsilon e, 33, 66 Einstein, A., 296 empty set, 2 equicontinuous, 145 equivalence classes, 2, 217, 236 Eudoxus,26 Euler, L., 53, 73, 96 Euler's constant, 113 expansion

binary, 31 continued fraction, 38 decimal,31

exterior differential, 287 exterior product, 277

(anti)commutativityof, 279 associativity of, 278

Fatou's lemma, 230, 248 Fermat, P., 121 Fibonacci numbers, 54 field,10-13

of sets, 221 ordered,12

finite induction, 4 finite intersection property, 151 finitely additive, 219

328 Index

fixed point theorem Brouwer, 305-307 contraction map, 137

Fourier, Jo, 121 Fourier coefficients, 164

uniqueness of, 165 Fourier series, 164 Fn'ichet, Mo, 153, 182, 221 Fn'ichet derivative, 182 Fubini, Go, 252 Fubini-Tonelli theorem, 245-247 function, 3, 17

absolute value, 17 Borel measurable, 244 characteristic, 225 complex, 96 continuous, 55, 57 convex, 71, 78, 96, 233, 250 Dirichlet, 56, 100 exponential, 65-67 graph of, 3, 251 greatest integer, 58 harmonic, 314 indicator, 225 inverse, 85 limit of, 56 logarithm, 67 lower semicontinuous, 151 measurable, 223

Borel,226 Lebesgue, 226

monotone, 56 nowhere differentiable contin-

uous,75 periodic, 162 piecewise linear, 62 polynomial, 17, 58 radial, 320 rational, 17, 58 sign, 17 simple, 225-228, 232 step, 232 trigonometric, 67-69 uniformly continuous, 61 upper semicontinuous, 70, 151

fundamental theorem of calculus, 110, 111

Gauss, CoFo, 54, 121, 267 geodesics, 161 gradient, 288 Gram determinant, 308 graph,256 Grassman, Ho, 283 Green, Go, 321 Green's identities, 314 group, 273

Hahn, Ho, 221 Halmos, Po, 26 Hardy, GoHo, 27 harmonic function, 314-318 Hausdorff, Fo, 153, 154, 211 Hausdorff

dimension, 211 measure, 210 space, 135

Heine, HoE., 72 Heine-Borel theorem, 140, 144,210 Hermite, Co, 27 Herstein, I., 27 Holder, 0o, 252 homeomorphic, 127 homeomorphism, 127 hypersurface, 265, 312

image, 3, 178 implicit function theorem, 194, 259

classical notation, 196 incommensurability, 8 increasing

function, 56 indexed family, 4 induced mapping of forms, 290-

291 inequality

arithmetic-geometric means, 234

Bernoulli, 30 Bessel, 168

Bunyakovsky, see Schwarz Cauchy, 129 Cauchy-Schwarz-Bunyakovsky,

235 Chebyshev, 233 Holder, 235 Jensen, 233 Liapounov, 234 Minkowski, 235 Schwarz, 108, 165

infimum, 15 injective, 3, 178 inner product, 129, 178, 271, 282 integers, 4 integrable, 230 integral

change of variables, 112 improper, 114 is a measure, 227, 228 linear change of variables, 241 linearity of, 227, 229 of differential form, 294 of form over manifold, 301 of nonnegative function, 228 of simple function, 227 Riemann, 98-114, 201, 231-

233 existence, 101-105

smooth change of variables, 241-244

integration by parts, 112, 247 interchange of limit and, 113 linearity of, 230 of series, 113

interior, 125 interior point, 124 intermediate value theorem, 59 intersection, 1 interval, 19

closed, 19 in Rn, 204 open, 19

intervals nested sequences, 19

Index 329

inverse function, 60, 191 inverse function theorem, 192, 257 isolated point, 125 isomorphism, 282

natural, 270

Jacobi, C.G.J., 252 Jacobian, 182, 294

matrix, 196 Jensen, J.W.L., 252 Jordan, C., 121

kernel, 178 Kolmogorov, A., 221 Kronecker, L., 54 Kronecker 6, 269, 276

multi-index, 271 Kronecker epsilon, 276, 311

L(JL),230 LP(JL) , 235

completeness of, 237 Lagrange, J.-1., 95, 96 Lagrange multipliers, 266 A r (alternating tensors of rank r),

272 'x-system, 237 Laplacian (Laplace operator), 314 largest integer in, 16 Lebesgue, H., 173, 221, 252 Leibniz, G., 53, 82, 95, 121 lemniscate, 258 level surface, 194 Levi, B., 228 Levi-Civita, T., 284 L'Hospital, G.-F.-A, 96 L'Hospital's rule, 86 limit point, 125 liminf, 34

of sets, 219 lim sup, 34

of sets, 219 Lindemann, F., 27 line segment, 185 linear algebra, 176-181

330 Index

linear functional, 269 linear transformation, 176, 240

metrics on, 179 Liouville, J., 20, 27, 219 Liouville's theorem

on algebraic numbers, 20 on harmonic functions, 317

Lipschitz condition, 138 local coordinate, 255 locally connected, 152 lower semicontinuous, 70, 159 lower sum, 100

manifold, 253, 254 construction of, 258-259 examples of, 256-258 orientable, 263 orientation of, 263 oriented, 263

map, 3 closed, 134 linear, 176 open, 134

mapping, see map matrix, 176-178 matrix product, 177 maximal,14 maximal rank, 259 maximum

existence of, 59, 142 local, 83, 184, 266

maximum principle, 316 Maxwell, J.e., 321 mean value inequality, 186 mean value theorem, 83, 185

for harmonic functions, 315 for integrals, 108

measurability criteria, 224 measurable

Lebesgue, 213, 215-218 measurable function, 223 measure, 206

Borel,206 counting, 203

determined by 1f-system, 238, 239

Hausdorff, 215, 321 inner, 220, 222 Lebesgue, 213, 215

Mertens, F., 53 metric, 127

product, 132 standard on R, 128 standard on R d , 130 uniform on C(X, Y), 145

metric space, 128-131, 214, 236 minimal, 14 minimum

existence of, 142 local, 83, 184, 266

Minkowski, H., 252 Mobius, A.F., 283 Mobius strip, 258, 267, 319 Monge, G., 267 monotone

convergence theorem, 228, 248 function, 56 sequence, 33, 64 set function, 203, 209

/-L* -measurable, 211 multinomial coefficient, 190 Miintz-Szasz theorem, 174

nabla (V'), 289 neighborhood, 56, 124 nested compact sets, 141 net, 153 Newton, I., 53, 95, 121 nonsingular, 178 norm, 129 normal derivative, 314 normal vector, 260 normal vector field, 312 null space, 178 numbers

algebraic, 20 complex, 25 extended real, 15, 203

operations, 223

natural, 2, 4 rational, 5

countability of, 7 real, 15

existence of, 22-23 uncountability of, 20

satanic, 221 transcendental, 20

open cover, 140 open set, 123, 215 operator norm, 179 orientable, 263

criterion for, 299 orientation, 262-265

positive of boundary, 264

standard, 265 usual, 263

oriented positively, 263

orthogonal transformation, 240 Osgood, W.F., 137, 153 Ostrogradski, M.V., 321 outer measure, 209

Hausdorff, 210, 221 Lebesgue, 210 Lebesgue-Stieltjes, 210, 221 metric, 214, 221

Parseval's relation, 169 partial derivative, 186

Leibniz notation, 197 multi-index notation, 190

partial derivatives, 191 higher order, 189 mixed, 189

partial order, 3 partition, 2, 98 partition of unity, 298, 299 path, 159

length of, 159 permutations, 273 Perron-Frobenius theorem, 319 'IT-system, 237

Index 331

Picard's theorem, 138 piecewise smooth, 168 Plato, 26 Poincare lemma, 291 point mass, 203 power series, 90

of matrices, 181 power set, 2 precompact, 144 primitive, 110 projection map, 133 pseudometric, 166, 220, 236

Rademacher functions, 167 radius of convergence, 90 Radon, J., 221 range, 3, 178 rectifiable, 159, 211 refinement, 98 relation, 2

equivalence, 2, 217 Ricci, G., 284 Riemann, B., 53, 121, 267 Riemann integral, see integral Riemann sum, 99 Riemann-Lebesgue lemma, 168 Riemannian structure, 293 Riesz, F., 252 Riesz-Fischer theorem, 237 Rogers, L.J., 252 Rolle, M., 96 root, existence of, 18 row vector, 183, 269 Russell, B., 27

Schwarz, H.A., 109 second axiom of countability, 131 Seidel, P., 73 selection, 98 separable, 126, 220 sequence, 4, 28

Cauchy, 36, 136 convergent, 28 divergent, 28 in topological space, 135

332 Index

monotone, 33 of functions, 62 of measures, 208 uniformly Cauchy, 63

series, 39 absolutely convergent, 41 alternating, 43 Cauchy product of, 49 comparison test, 40 conditionally convergent, 41 engineers test, 40 for e, 42 Fourier, 122, 164 geometric, 41 harmonic, 30, 41, 42 partial sum, 39 ratio test, 42 rearrangement of, 45 root test, 41 telescoping, 41 trigonometric, 174 unordered, 47

set, 1 Borel, 205, 214, 215, 220, 224 closed, 124 convex, 96, 152 directed, 53 finite, 6 infinite, 6 Lebesgue measurable, 248 unmeasurable, 217

Sierpinski, W., 154, 174 a-algebra, 204, 238

generated by, 205 a-field, 221 sign of permutation, 273 simple function, 225

canonical representation, 225 smooth, 254, 255

forms, 286 space

door, 150 Euclidean, 123 Hausdorff, 135 Hilbert, 130

metric, 128 topological, 123

square root algorithm, 31 Stokes, G., 73, 321 Stokes' theorem, 303

classical case, 319 Stone, M., 158, 174 Stone's theorem, 157 Stone-Weierstrass theorem, 158 subgroup, 218 subsequence, 34 subspace, 131 summable, 230 summation by parts, 44 summation convention, 281 support line, 81 supremum, 15 surface, 253, 257 surjective, 3, 178 symmetric tensor, 282

TT (tensors of rank r), 270 tangent bundle, 260 tangent space, 253, 260 tangent vector, 260

characterization of, 261, 262 Taylor, B., 96 Taylor polynomial, 89, 191 Taylor's theorem, 89, 96, 120

in n variables, 190 tensor

alternating, 272 covariant, 270 elementary, 283 rank of, 270

tensor field, 285 of class C k , 286

tensor product, 271 tensors of rank r

basis for, 271 Thomson, W. (Kelvin), 321 Toeplitz, 0., 26 Tonelli, L., 252 topological space, 220 topologies

partial ordering of, 124 topology, 123

base for, 131 discrete, 124 metric, 128 of uniform convergence, 232 product, 132, 134 relative, 131 stronger, finer, 124 trivial, 124 uniform on C(X, Y), 145 usual of R, 124 weaker, coarser, 124

torus, 257 dense line in, 265 dense line on, 266

total order, 3 totally bounded, 142 trace norm, 179 transition function, 256 translation invariant, 217 translation operator, 97 transpose, 178 transposition, 274 trigonometric polynomials, 162 triple scalar product, 279 Tychonoff, A., 153

uncountable, 6

Index 333

uniform continuity, 142 uniform metric, 145 uniform topology, 145 uniformly continuous, 131 union, 1 unmeasurable set, 218 unordered sums, 208 upper halfspace, 254 upper sum, 100 Urysohn, P., 153

Van Vleck, E.B., 222 vector field, 260, 288

normal, 265 Vitali, G., 222 volume, 307-311

Walsh functions, 172 wedge product, see exterior prod­

uct Weierstrass, K., 27, 72, 73, 153,

173 Weierstrass

M-test, 65 approximation theorem, 155

well separated, 214 well-ordered, 4 Weyl, H., 170

Undergraduate Texts in Mathematics (continued/rom page ii)

HiimmerlinIHoffmann: Numerical Mathematics. Readings in Mathematics.

HarrisIHirstJMossinghoff: Combinatorics and Graph Theory.

Hartshorne: Geometry: Euclid and Beyond.

Hijab: Introduction to Calculus and Classical Analysis.

HiltonlHoltonlPedersen: Mathematical Reflections: In a Room with Many Mirrors.

Iooss/Joseph: Elementary Stability and Bifurcation Theory. Second edition.

Isaac: The Pleasures of Probability. Readings in Mathematics.

James: Topological and Uniform Spaces.

Jiinich: Linear Algebra. Jiinich: Topology. Jiinich: Vector Analysis. Kemeny/Snell: Finite Markov Chains. Kinsey: Topology of Surfaces. KIambauer: Aspects of Calculus. Lang: A First Course in Calculus. Fifth

edition. Lang: Calculus of Several Variables.

Third edition. Lang: Introduction to Linear Algebra.

Second edition. Lang: Linear Algebra. Third edition. Lang: Undergraduate Algebra. Second

edition. Lang: Undergraduate Analysis. LaxlBursteinlLax: Calculus with

Applications and Computing. Volume 1.

LeCuyer: College Mathematics with APL.

LidllPilz: Applied Abstract Algebra. Second edition.

Logan: Applied Partial Differential Equations.

Macki-Strauss: Introduction to Optimal Control Theory.

Malitz: Introduction to Mathematical Logic.

MarsdenlWeinstein: Calculus I, II, III. Second edition.

Martin: The Foundations of Geometry and the Non-Euclidean Plane.

Martin: Geometric Constructions. Martin: Transformation Geometry: An

Introduction to Symmetry. MilImanlParker: Geometry: A Metric

Approach with Models. Second edition.

Moschovakis: Notes on Set Theory. Owen: A First Course in the

Mathematical Foundations of Thermodynamics.

Palka: An Introduction to Complex Function Theory.

Pedrick: A First Course in Analysis. PeressinilSullivanlUhl: The Mathematics

of Nonlinear Programming. PrenowitzlJantosciak: Join Geometries. Priestley: Calculus: A Liberal Art.

Second edition. ProtterlMorrey: A First Course in Real

Analysis. Second edition. ProtterlMorrey: Intermediate Calculus.

Second edition. Roman: An Introduction to Coding and

Information Theory. Ross: Elementary Analysis: The Theory

of Calculus. Samuel: Projective Geometry.

Readings in Mathematics. Scharlau/Opolka: From Fermat to

Minkowski. Schiff: The Laplace Transform: Theory

and Applications. Sethuraman: Rings, Fields, and Vector

Spaces: An Approach to Geometric Constructability .

Sigler: Algebra. Silvermanrrate: Rational Points on

Elliptic Curves. Simmonds: A Brief on Tensor Analysis.

Second edition.

Undergraduate Texts in Mathematics

Singer: Geometry: Plane and Fancy. Singerffhorpe: Lecture Notes on

Elementary Topology and Geometry.

Smith: Linear Algebra. Third edition. Smith: Primer of Modern Analysis.

Second edition. StantonlWhite: Constructive

Combinatorics. Stillwell: Elements of Algebra:

Geometry, Numbers, Equations. Stillwell: Mathematics and Its History. Stillwell: Numbers and Geometry.

Readings in Mathematics.

Strayer: Linear Programming and Its Applications.

Thorpe: Elementary Topics in Differential Geometry.

Toth: Glimpses of Algebra and Geometry. Readings in Mathematics.

Troutman: Variational Calculus and Optimal Control. Second edition.

Valenza: Linear Algebra: An Introduction to Abstract Mathematics.

WbyburnlDuda: Dynamic Topology. Wilson: Much Ado About Calculus.