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cohomologous transition functions, 288, 380
“, 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

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