“, locally Lipschitzian, 9

cohomology algebra, De Rham, 354

“, smooth, 9

“ classes of transition functions, 288

“, topologically real analytic, 99

co-idempotent, 247

colimit, 576

D

commensurable groups, 510

δ, natural embedding into the bidual, 16

commutative algebra, 57

d, di¬erentiation operator, 33

comp, the composition mapping, 31

dn , iterated directional derivative, 26

compact-open C k -topology, 436 v

(d) d [j]

Da E := j=1 D , space of operational

“ topology, 434

tangent vectors of order ¤ d, 278

compatible vector bundle charts, 287

(D(k) ) M , operational cotangent bundle, 337

completant set, bounded, 580

Index 613

[d]

Da E, space of operational tangent vectors of Dunford-Pettis property, 200

homogeneous order d, 278

E

De Rham cohomology algebra, 354

E , space of bounded linear functionals, 8

de¬nite integral, 16

E — , dual space of continuous linear

density of subset of N, 22

functionals, 8

“ number densX of a topological space, 152

EB , linear space generated by B ⊆ E, 11, 576

dentable subset, 135

Der(C ∞ (M, R)), space of operational vector Eborn , bornologi¬cation of E, 35

ˆ

¬elds, 322 E, completion of E, 16

derivation over eva , continuous, 276 embedding of manifolds, 269

“, algebraic, 358 “, bornological, 48

“, order of a, 277 equicontinuous sets, 15

“, graded, 358 equidistant di¬erence quotient, 119

derivative of a curve, 8 equivalent atlas, 264

“, covariant, 397 evolution, right, 410

“, covariant exterior, 392, 399 exact contact structure, 467

“, covariant Lie, 399 “ di¬erential forms, 353

“, directional, 128 expansion at x, 311

“, left logarithmic, 404 “ property, 311

“, left trivialized, 374 exponential law, 445

“, Lie, 347, 360 “ mapping, 372

“, n-th, 58 expose a subset, 130

“, right logarithmic, 404 extension of groups, 412

“, unidirectional iterated, 62 “ property, 47

di¬eomorphic manifolds, 264 “ property, scalar valued, 221

di¬eomorphism, contact, 467 “ property, vector valued, 221

“ group, 454 “, k-jet, 431

exterior algebra, 57

“, F -foliated, 272

“ derivative, covariant, 392, 399

“, symplectic, 460

“ derivative, global formula for, 342

“, holomorphic, 264

“, real analytic, 264

F

di¬erence quotient, 13, 119

f -dependent, 366

“ quotient, equidistant, 119

f -related vector ¬elds, 329

di¬erentiable curve, 8

“ vector valued di¬erential forms, 366

di¬erential forms, 353

F -evaluating, 184

“ forms, f -related vector valued, 366

F -foliated di¬eomorphism, 272

“ forms, horizontal, 392

fast converging sequence, 17, 18

“ forms, horizontal G-equivariant W -valued,

“ falling, 17

401

¬ber bundle, 375

“ forms, vector valued kinematic, 359

“ bundle, gauge group of a, 479

“ group of order k, 432

“ bundle, principal, 380

“ of a function, 285

“ of the operational tangent bundle, 284

di¬erentiation operator, 33

“ of the tangent bundle, 284

direct sum, 576

¬bered composition of jets, 431

directed set, 577

¬nal smooth mapping, 272

directional derivative, 128

¬nite type polynomial, 60

distinguished chart of a foliation, 273

¬rst uncountable ordinal number ω 1 , 49

dual conjugate pair, 586

“ mapping — , 8 ¬‚at at 0, in¬nitely, 61

¬‚atness, order of, 539

“ of a convex function, 131

¬‚ip, canonical, 293

“ of a locally convex space, strong, 579

¬‚ow line of a kinematic vector ¬eld, 329

“ pair, weak topology for a, 578

“ of a kinematic vector ¬eld, local, 331

“ space E of bounded linear functionals on a

foliation, 272

space E, 8

“ space E — of continuous linear functionals on foot point projection, 284

a space E, 8 formally real commutative algebra, 305

614 Index

frame bundle, 477 homogeneous operational tangent vector of

order d, 277

“ bundle, nonlinear, 477

Fr´chet space, 577

e homomorphism of G-bundles, 384

“ space, graded, 557 “ of principal ¬ber bundles, 381

“ space, tame graded, 559 “ of vector bundles, 289

“-di¬erentiable, 128 “ over ¦ of principal bundles, 381

Frobenius theorem, 330, 331 homotopy operator, 355

Fr¨licher space, 238

o horizontal bundle, 376

“-Nijenhuis bracket, 361 “ di¬erential forms, 392

functor, smooth, 290 “ G-equivariant W -valued di¬erential forms,

fundamental theorem of calculus, 17 401

“ vector ¬eld, 375, 375 “ lift, 376

“ projection, 376

G “ space of a connection, 366

G-atlas, 379

“ vectors of a ¬ber bundle, 376

G-bundle, 379

G-bundle, homomorphism of, 384

I

G-structure, 379

induced connection, 394, 394

GL(k, ∞; R), Stiefel manifold of k-frames, 514

inductive limit, 577

Gˆteaux-di¬erentiable, 128

a

in¬nite polygon, 18

gauge group of a ¬ber bundle, 479

in¬nitely ¬‚at at 0, 61

“ transformations, 385

initial mapping, 268

general curve lemma, 118

inner automorphism, 373

generating set of functions for a Fr¨licher

o

insertion operator, 341, 399

space, 239

integral curve of a kinematic vector ¬eld, 329

germ of f along A, 274

“ mapping, 136

germs along A of holomorphic functions, 92

“, de¬nite, 16

global resolvent set, 549

“, Riemann, 15

globally Hamiltonian vector ¬elds, 460

m

interpolation polynomial P(t ,...,t ) , 228

graded derivations, 358 m

0

graded Fr´chet space, 557

e invariant kinematic vector ¬eld, 370

“ Fr´chet space, tame, 559

e involution, canonical, 293

“-commutative algebra, 57 isomorphism, bornological, 8

graph topology, 435 “ of vector bundles, 289

Grassmann manifold G(k, ∞; R), 514

group, di¬eomorphism, 454 J

“, holonomy, 426 jets, 431

“, Lie, 369

“, reduction of the structure, 381 K

“, regular Lie, 410

k-jet extension, 431

“, restricted holonomy, 426

k-jets, 431

“, smooth, 432

kE, 37

groups, extension of, 412

K , set of accumulation points of K, 143

Kelley-¬cation, 37

H

Killing form on gl(∞), 520

H(U, F ), 90

kinematic 1-form, 337

Hamiltonian vector ¬eld, 460

“ cotangent bundle, 337

Hausdor¬, smoothly, 265

“ di¬erential forms, vector valued, 359

H¨lder mapping, 128

o

“ tangent bundle, 284

holomorphic atlas, 264

“ tangent vector, 276

“ curve, 81

“ vector ¬eld, 321

“ di¬eomorphisms, 264

“ vector ¬eld, ¬‚ow line of a, 329

“ mapping, 83

“ vector ¬eld, left invariant, 370

“ mappings, initial, 268

“ vector ¬eld, local ¬‚ow of a, 331

“ vector bundle, 287

holonomy group, 426 K¨the sequence space, 71, 581

o

Index 615

L “ convex space, ultrabornologi¬cation of a, 575

—, adjoint mapping, 8 “ convex space, weakly realcompact, 196

∞ (X, F ), 21 “ convex topology, bornologi¬cation of a, 13

1 (X), 50 “ convex vector space, ultra-bornological, 580

L(E, F ), 33 locally Hamiltonian vector ¬eld, 460

—

L(Eequi , R), 15 “ Lipschitzian curve, 9

“ uniformly rotund norm, 147

L(E1 , ..., En ; F ), 53

Lipk -curve, 9 logarithmic derivative, left or right, 404

Lipk -mapping, 118

M

Lipk (A, E), space of functions with locally

ext

m-evaluating, 184

bounded di¬erence quotients, 229

Lipk , space of C k -functions with global m-small zerosets, 205

K

MC (complexi¬cation of M ), complex

Lipschitz-constant K for the k-th

manifold, 105

derivatives, 159

Lipk k µ-converging sequence, 35

global , space of C -functions with k-th

M -convergence condition, 39

derivatives globally Lipschitz, 159

M -convergent net, 12

Lagrange submanifold, 460

M -converging sequence, 12

leaf of a foliation, 273

Mackey adherence, 48, 51

left invariant kinematic vector ¬eld, 370

“ adherence of order ±, 49

“ logarithmic derivative, 404

“ approximation property, 70

“ Maurer-Cartan form, 406

“ complete space, 15

“ trivialized derivative, 374

“ convergent net, 12

Legendre mapping, 468

“ convergent sequence, 12

“ submanifold, 468

“, second countability condition of, 159

Leibniz formula, 54

“-Cauchy net, 14

Lie bracket of vector ¬elds, 324

“-closure topology, 19

“ derivative, 347, 360

Mackey™s countability condition, 236

“ derivative, covariant, 399

manifold, 264

“ group, 369

“ MC (complexi¬cation of M ), complex, 105

“ group, regular, 410

“ structure of C∞ (M, N ), 439

lift, horizontal, 376

“, contact, 467

“, vertical, 293

“, natural topology on a, 265

limit, 577

“, pure, 265

“, inductive, 577

“, symplectic, 460

“, projective, 577

mapping, bornological, 19

linear connection, 396, 397

“, bounded, 19

“ mapping, bounded, 8

“, tame smooth, 563

Liouville form, 523

“ between Fr¨licher spaces, smooth, 239

o

Lipschitz bound, absolutely convex, 17

“, 1-homogeneous, 34

“ condition, 9

“, biholomorphic, 264

“ mapping, 128

“, bounded linear, 8

Lipschitzian curve, locally, 9

“, carrier of a, 153

local addition, 441

“, complex di¬erentiable, 81

“ ¬‚ow of a kinematic vector ¬eld, 331

“, exponential, 372

locally complete space, 20

“, ¬nal, 272

locally convex space, 575

“, H¨lder, 128

o

“ convex space, barrelled, 579

“, holomorphic, 83

“ convex space, bornological, 575

“, initial, 268

“ convex space, bornologi¬cation of a, 575

“, integral, 136

“ convex space, bornology of a, 8, 575

“, Legendre, 468

“ convex space, completion of a, 16

“, Lipschitz, 128

“ convex space, nuclear, 580

“, nuclear, 136

“ convex space, re¬‚exive, 579

“, proper, 445

“ convex space, Schwartz, 579

“, real analytic, 102

“ convex space, strong dual of a, 579

“ convex space, strongly nuclear, 580 “, smooth, 30

616 Index

“, support of a, 153 “ tangent vector of order d, homogeneous, 277

“ vector ¬eld, 321

“, transposed, 326

operator, di¬erentiation, 33

“, zero set of a, 153

“, homotopy, 355

Maurer-Cartan form, 373

“, insertion, 341, 399

“ formula, 378

“, nuclear, 580

maximal atlas, 264

“, strongly nuclear, 580

mean value theorem, 10

“, trace class, 580

mesh of a partition, 15

“, trace of an, 580

Minkowski functional, 11, 575

order of a derivation, 277

modeling convenient vector spaces of a

“ of ¬‚atness, 539

manifold, 265

ordinal number ω 1 , ¬rst uncountable, 49

modular 1-form, 337

modules, bounded, 63

P

monomial of degree p, 60

m

P(t ,...,t ) , interpolation polynomial, 228

Montel space, 579 m

0

Polyp (E, F ), space of polynomials of degree

multiplicity, 539

¤ p, 61

N paracompact, smoothly, 165

n-th derivative, 58 parallel transport on a ¬ber bundle, 378

n-transitive action, 472 partition of unity, 165

natural bilinear concomitants, 367 plaque of a foliation, 273

“ topology, 488 Poincar´ lemma, relative, 461

e

“ topology on a manifold, 265 “ lemma, 350

polar U o of a set, 578

net, Mackey convergent, 12

“, Mackey-Cauchy, 14 polynomial, 60

“, M -convergent, 12 “, ¬nite type, 60

Nijenhuis tensor, 368 power series space of in¬nite type, 72

“-Richardson bracket, 359 precompact, 576

nonlinear frame bundle of a ¬ber bundle, 477 PRI, projective resolution of identity on a

norm, locally uniformly rotund, 147 Banach space, 588

“, rough, 135 principal bundle, 380

“, strongly rough, 158 “ bundle of embeddings, 474

“, uniformly convex, 204 “ connection, 387

normal bundle, 438 “ right action, 380

“, smoothly, 165 product of manifolds, 264

norming pair, 582 “ rule, 54

nuclear locally convex space, 580 projection of a ¬ber bundle, 376

“ mapping, 136 “ of a vector bundle, 287

“ operator, 580 “, foot point, 284

“, horizontal, 376

O “, vertical, 293, 376

O(k, ∞; R), Stiefel manifold of orthonormal projective generator, 584

k-frames, 514 “ limit, 577

k (M ), space of di¬erential forms, 352

„¦ “ resolution of identity, 588

„¦k (M, V ), space of di¬erential forms with “ resolution of identity, separable, 588

values in a convenient vector space V , 352 proper mapping, 445

k (M ; E), space of di¬erential forms with

„¦ pseudo-isotopic di¬eomorphisms, 510

values in a vector bundle E, 352 pullback, 377

ω1 , ¬rst uncountable ordinal number, 49 “ of vector bundles, 290

ω-isolating, 203 pure manifold, 265

one parameter subgroup, 371

R

operational 1-form, 337

“ 1-forms of order ¤ k, 337 R(c, Z, ξ), Riemann sum, 15

“ cotangent bundle, 337 radial set, 35

“ tangent bundle, 283 Radon-Nikodym property of a bounded convex

“ tangent vector, 276 subset of a Banach space, 135

Index 617

real analytic atlas, 264 sequential adherence, 41

“ analytic curve, bornologically, 99 Silva space, 171, 581

“ analytic curve, topologically, 99 slice, 480, 480

“ analytic di¬eomorphisms, 264 smooth atlas, 264

“ analytic mapping, 102 “ curve, 9

“ curves in C∞ (M, N ), 442

“ analytic mapping, initial, 268

“ function of class S, 153

“ analytic vector bundle, 287

realcompact locally convex space, weakly, 196 “ functor, 290