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“, 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
“, smoothly, 184 “ group, 432
reduction of the structure group, 381 “ mapping, 30
re¬‚exive convenient vector space, 20 “ mapping between Fr¨licher spaces, 239
o
“ locally convex space, 579 “ mapping, ¬nal, 272
regular Lie group, 410 “ mapping, initial, 268
“, completely, 46 “ mapping, tame, 563
“, smoothly, 153 “ seminorm, 129
relative Poincar´ lemma, 461
e “ structure, 238
representation, 528 smoothly Hausdor¬, 265
resolution of identity, projective, 588 “ normal space, 165
resolvent set, global, 549 “ paracompact space, 165
restricted holonomy group, 426 “ realcompact space, 184
Riemann integral, 15 “ regular space, 153
“ sum, 15 smoothness of composition, 444
right action, principal, 380 space of bounded linear mappings, 33
“ evolution, 410 “ of bounded n-linear mappings, 53
“ invariant kinematic vector ¬eld, 370 “ of holomorphic functions, 91
“ logarithmic derivative, 404 “ of holomorphic mappings, 90
rotund norm, locally uniformly, 147 “ of real analytic curves, 102
rough norm, 135 “ of real analytic mappings, 102
“ norm, strongly, 158 “ of smooth mappings, 30
“ of connections, 479
S “ of germs of real analytic functions, 105
Sn , group of permutations, 57 “ of real analytic functions, 105
S-boundedness principle, uniform, 65 special curve lemma, 18
S-functions, 153 splitting submanifold, 268
S-normal space, 165 SPRI, separable projective resolution of
S-paracompact space, 165 identity, 588
S-partition of unity, 165 standard ¬ber of a ¬ber bundle, 376
S-regular space, 153 “ ¬ber of a vector bundle, 287
sE sequentially generated topology on E, 37 stereographic atlas, 512
Stiefel manifold GL(k, ∞; R) of k-frames, 514
scalar valued extension property, 221
“ manifold O(k, ∞; R) of orthonormal k-
scalarly true property, 11
frames, 514
scattered topological space, 146
strict inductive limit, 577
Schwartz locally convex space, 579
strong dual of a locally convex space, 579
second countability condition of Mackey, 159
“ operator topology, 528
“ countable, has countable base of topology,
296 “ symplectic structure, 523
section of a vector bundle, 294 strongly expose a subset, 130
seminorm, 575 “ nuclear locally convex space, 580
“, smooth, 129 “ nuclear operator, 580
separable topological space, 578 “ rough norm, 158
“ projective resolution of identity, 588 submanifold, 268
sequence space, K¨the, 71, 581
o “ charts, 268
“, fast converging, 17, 18 “, Lagrange, 460
“, Mackey convergent, 12 “, Legendre, 468
“, M -converging, 35 “, splitting, 268
“, µ-converging, 35 subordinated partition of unity, 165
618

super-re¬‚exive Banach space, 204 “ vector bundle, 522
support of a mapping, 153
V
“ of a section, 294
symmetric algebra, 57 Valdivia compact space, 591
symmetrizer sym, 57 Vandermonde™s determinant, 27
symplectic di¬eomorphism, 460 vector bundle, 287
“ form, 460 “ bundle, universal, 522
“ manifold, 460 “ ¬eld, characteristic, 467
“ structure, strong, 523 “ ¬eld, ¬‚ow line of a kinematic, 329
“ structure, weak, 523 “ ¬eld, fundamental, 375, 375
“ vector ¬eld, 460 “ ¬eld, globally Hamiltonian, 460
symplectomorphism, 460 “ ¬eld, integral curve of a kinematic, 329
“ ¬eld, kinematic, 321
T “ ¬eld, left invariant kinematic, 370
tame equivalent gradings of degree r and base “ ¬eld, local ¬‚ow of a kinematic, 331
b, 557 “ ¬eld, locally Hamiltonian, 460
“ graded Fr´chet space, 559
e “ ¬eld, operational, 321
“ linear mapping of degree d and base b, 557 “ ¬eld, right invariant kinematic, 370
“ non-linear mapping, 560 “ ¬eld, symplectic, 460
“ smooth map, 563 “ ¬elds, f -related, 329
tangent bundle, kinematic, 284 “ ¬elds, Lie bracket of, 324
“ bundle, operational, 283 vector space, arc-generated, 39
“ hyperplane, 130 “ space, convenient, 2, 7, 20
“ vector, kinematic, 276 “ valued extension property, 221
“ vector, operational, 276 “ valued kinematic di¬erential forms, 359
tensor algebra, 57 vertical bundle, 292
“ product, bornological, 55 “ bundle of a ¬ber bundle, 376
topologically real analytic curve, 99 “ lift, 293
topology on a manifold, natural, 265 “ projection, 293, 376
“, compact-open, 434 “ space of a connection, 366
“, graph, 435
W
“, Mackey-closure, 19
“, natural, 488 WCD, weakly countably determined space,
“, strong operator, 528 585
“, wholly open, 435 WCG, weakly compactly generated space, 135
trace class operator, 580 weak symplectic structure, 523
“ of an operator, 580 “ topology for a dual pair, 578
transition function for vector bundle charts, weakly Asplund space, 136
287 “ realcompact locally convex space, 196
“ functions of a ¬ber bundle, 376 Weil algebra, 306
transposed mapping, 326 “ functor, 307, 309
Whitney C k -topology, 436
truncated composition, 431
tubular neighborhood, 438 wholly open topology, 435
WO-topology, 435
WO0 -topology, 435
U
U o , polar, 578 WOk -topology, 436
ultrabornological locally convex space, 580
X
ultrabornologi¬cation, 575
X(M), space of kinematic vector ¬elds, 321
unidirectional iterated derivative, 62
uniform boundedness principle, 61
Z
“ S-boundedness principle, 65
uniformly convex norm, 204 zero section, 293
universal covering space, 271 “ set of a mapping, 153

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