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