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Kriegl A., Michor P.W. — The Convenient Setting of Global Analysis
Kriegl A., Michor P.W. — The Convenient Setting of Global Analysis

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Название: The Convenient Setting of Global Analysis

Авторы: Kriegl A., Michor P.W.

Аннотация:

This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Fréchet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.


Язык: en

Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1997

Количество страниц: 618

Добавлена в каталог: 02.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(D^{(k)})'M$, operational cotangent bundle      337
$A^{<k>}$, set of points in $A^{k+1}$ with pairwise distinct coordinates      227
$B_{\varepsilon}(x)$, $\varepsilon$-ball centered at x      156
$C^k$-topology, compact-open      436
$C^k$-topology, Whitney      436
$C^k_B$ space of $C^k$-functions with k-th derivative bounded by b      159
$C^k_b$, space of $C^k$-functions with k-th derivative bounded      159
$C^{\infty}$, smooth      30
$C^{\infty}$-algebra, augmented local convenient      316
$C^{\infty}$-algebras, Chart description of functors induced by      316
$c^{\infty}$-approximation property      70
$c^{\infty}$-complete space      20
$c^{\infty}$-completion      47
$c^{\infty}$-open set      19
$C^{\infty}$-structure      264
$c^{\infty}$-topology      19
$C^{\infty}(u, f)$, space of smooth mappings      30
$C^{\infty}(\mathbb{R}, E)$, space of smooth curves      28
$C^{\infty}_b$, space of smooth functions with bounded derivatives      159
$C^{\omega}$-manifold structure of $C^{\omega}(M,N)$      442
$C^{\omega}$-manifold structure on $\mathfrak{C}^{\infty}(m,n)$      443
$c_0$-ext, a class of locally convex spaces      212
$c_0(X)$      50
$c_0(\Gamma)$, space of 0-sequences      142
$c_c(x)$      50
$Der(C^{\infty}(M, \mathbb{R}))$, space of operational vector fields      322
$D_a^{(d)}E :=\bigoplus^d_{j=1} D^{[j]}$, space of operational tangent vectors of order $\leq$ d      278
$D_a^{[d]}E$, space of operational tangent vectors of homogeneous order d      278
$d_v^n$, iterated directional derivative      26
$E_B$, linear space generated by B C E      11 576
$E_{born}$, bornologification of E      35
$GL(k,\infty; \mathbb{R})$, Stiefel manifold of k-frames      514
$L(E^*_{equi},\mathbb{R})$      15
$L(E_1 ,...,E_n; F)$      53
$l^*$, adjoint mapping      8
$l^1(X)$      50
$l^{\infty}(X, F)$      21
$M_C$ (complexification of m), complex manifold      105
$O(k,\infty;\mathbb{R})$, Stiefel manifold of orthonormal k-frames      514
$Poly^p(E, F)$, space of polynomials of degree $\leq$ p      61
$P^m_{(t_0,...,t_m)}$, interpolation polynomial      228
$R(c,Z,\xi)$, Riemann sum      15
$S_n$, group of permutations      57
$U^0$, polar      578
$WO^0$-topology      435
$WO^k$-topology      436
$\bigotimes_\beta$, bornological tensor algebra      57
$\bigvee$, bornological symmetric algebra      57
$\bigwedge$, bornological exterior algebra      57
$\Delta$, diagonal mapping      59
$\delta$, natural embedding into the bidual      16
$\mathcal{H}(U, F)$      90
$\mathcal{L}ip^k$-curve      9
$\mathcal{L}ip^k$-mapping      118
$\mathcal{L}ip^k_K$, space of $C^k$-functions with global Lipschitz-constant K for the k-th derivatives      159
$\mathcal{L}ip^k_{ext}(A, E)$, space of functions with locally bounded difference quotients      229
$\mathcal{L}ip^k_{global}$, space of $C^k$-functions with k-th derivatives globally Lipschitz      159
$\mathfrak{C}^{\infty}(m,n)$, manifold of smooth mappings      439
$\mathfrak{N}_0$, first countable cardinal      46
$\mu$-converging sequence      35
$\omega$-isolating      203
$\Omega^k(M)$, space of differential forms      352
$\Omega^k(M, V)$, space of differential forms with values in a convenient vector space V      352
$\Omega^k(M; E)$, space of differential forms with values in a vector bundle E      352
$\omega_1$, first uncountable ordinal number      49
$\otimes_\beta$ bornological tensor product      55
$||\quad||_1$ 1-norm      137
$||\quad||_{\infty}$, $infty$-norm      139
1-form, kinematic      337
1-form, modular      337
1-form, operational      337
1-isolating      203
Absolutely convex      576
Absolutely convex Lipschitz bound      17
Absorbing      575
Absorbs      34
Addition, local      441
Adherence of order $\alpha$, Mackey      48 49 51
Adherence of order $\alpha$, sequential      41
Adjoint mapping $l^*$      8
Adjoint mapping $l^*$, representation      373
Algebra, bounded      63
Algebra, commutative      57
Algebra, De Rham cohomology      354
Algebra, exterior      57
Algebra, formally real commutative      305
Algebra, graded-commutative      57
Algebra, symmetric      57
Algebra, tensor      57
Algebra, Weil      306
Algebraic bracket of vector valued differential forms      359
Algebraic derivation      358
Almost complex structure      368
Almost continuous function      87
Alternating tensor      57
Alternator, alt      57
Analytic subsets      241
Anti-derivative      20
Approximation of unity      27
Approximation property, bornological      70 280
Approximation property, Mackey      70
arc-generated vector space      39
Asplund space      135
Asplund space, weakly      136
Associated bundle      382
Atlas      264
Atlas, equivalent      264
Atlas, holomorphic      264
Atlas, principal bundle      380
Atlas, real analytic      264
Atlas, smooth      264
Atlas, stereographic      512
Atlas, vector bundle      287
Augmented local convenient $C^{\infty}$-algebra      316
Automorphism, inner      373
Barrelled locally convex space      579
Base of a vector bundle      287
Base space of a fiber bundle      376
Basis of a fiber bundle      376
Bezoutiant matrix      537
Bianchi identity      377
Biholomorphic mappings      264
Bilinear concomitants, natural      367
Bipolar $U^{\infty}$      16
Bornivorous      34
Bornivorous, set      35
Bornological approximation property      70 280
Bornological embedding      48
Bornological isomorphism      8
Bornological locally convex space      575
Bornological mapping      19
Bornological tensor product $\otimes_\beta$      55
Bornological vector space      34
Bornologically compact set      62 88
Bornologically compact subset      41
Bornologically real analytic curve      99
Bornologification      35
Bornologification of a locally convex space      575
Bornology of a locally convex space      8 575
Bornology on a set      21
Bounded algebra      63
Bounded completant set      580
Bounded linear mapping      8
Bounded mapping      19
Bounded modules      63
Bounded set      575
Bounding set      19
Bump function      153
Canonical flip      293
Canonical involution      293
Carrier of a mapping      153
Cartan developing      427
Cartesian closedness      30
Cech cohomology set      288
Chain rule      33
Characteristic vector field      467
Chart changing mapping      264
Chart description of functors induced by $C^{\infty}$-algebras      316
Chart description of Weil functors      307
Chart of a foliation, distinguished      273
Chart, submanifold      268
Chart, vector bundle      287
Christoffel forms      377
Classifying connection      485
Classifying space      485 487
Closed differential forms      353
Co-algebras, convenient      246
Co-commutative      246
Co-idempotent      247
CO-topology      434 436
Cocurvature of a connection      366
Cocycle condition      288 376 414
Cocycle of transition functions      288 376
Cohomologous transition functions      288 380
Cohomology algebra, De Rham      354
Cohomology classes of transition functions      288
Colimit      576
Commensurable groups      510
Commutative algebra      57
Comp, the composition mapping      31
Compact-open $C^k$-topology      436
Compact-open topology      434
Compatible vector bundle charts      287
Completant set, bounded      580
Complete space, locally      20
Complete space, Mackey      15
Completely regular space      46
Completion of a locally convex space      16
Complex differentiable mapping      81
Complexification $M_C$ of manifold M      20
Composition, smoothness of      444
Composition, truncated      431
Conjugate pair      583
Conjugate pair, dual      586
Conjugation      373
Connection      366
Connection form, Lie algebra valued      387
Connection on a fiber bundle      376
Connection, classifying      485
Connection, induced      394 394
Connection, linear      396 397
Connection, principal      387
Connections, space of      479
Connector      397
Contact diffeomorphisms      467
Contact distribution      467
Contact form      467
Contact graph of a diffeomorphism      470
Contact manifold      467
Contact structure, exact      467
Continuous derivation over $ev_a$      276
Convenient $C^{\infty}$-algebra, augmented local      316
Convenient co-algebras      246
Convenient vector space      2 7 20
Convex function, dual of a      131
Convolution      27
Coproduct      576
Cotangent bundle, kinematic      337
Cotangent bundle, operational      337
Covariant derivative      397
Covariant exterior derivative      392 399
Covariant Lie derivative      399
Covering space, universal      271
Curvature      366 398
Curve, bornologically real analytic      99
Curve, differentiable      8
Curve, holomorphic      81
Curve, locally Lipschitzian      9
Curve, smooth      9
Curve, topologically real analytic      99
d, differentiation operator      33
De Rham cohomology algebra      354
Definite integral      16
Density number densX of a topological space      152
Density of subset of $\mathbb{N}$      22
Dentable subset      135
Derivation over $ev_a$, continuous      276
Derivation, algebraic      358
Derivation, graded      358
Derivation, order of a      277
Derivative of a curve      8
Derivative, covariant      397
Derivative, covariant exterior      392 399
Derivative, covariant Lie      399
Derivative, directional      128
Derivative, left logarithmic      404
Derivative, left trivialized      374
Derivative, Lie      347 360
Derivative, n-th      58
Derivative, right logarithmic      404
Derivative, unidirectional iterated      62
Diffeomorphic manifolds      264
Diffeomorphism, contact      467
Diffeomorphism, F-foliated      272
Diffeomorphism, group      454
Diffeomorphism, holomorphic      264
Diffeomorphism, real analytic      264
Diffeomorphism, symplectic      460
Difference quotient      13 119
Difference quotient, equidistant      119
Differentiable curve      8
Differential forms      353
Differential forms, f-related vector valued      366
Differential forms, horizontal      392
Differential forms, horizontal G-equivariant W-valued      401
Differential forms, vector valued kinematic      359
Differential group of order k      432
Differential of a function      285
Differentiation operator      33
Direct sum      576
Directed set      577
Directional derivative      128
Distinguished chart of a foliation      273
Dual conjugate pair      586
Dual mapping $l^*$      8
Dual of a convex function      131
Dual of a locally convex space, strong      579
Dual pair, weak topology for a      578
Dual space E' of bounded linear functionals on a space E      8
Dual space E* of continuous linear functionals on a space E      8
Dunford — Pettis property      200
E*, dual space of continuous linear functionals      8
E, completion of E      16
Embedding of manifolds      269
Embedding, bornological      48
Equicontinuous sets      15
Equidistant difference quotient      119
Equivalent atlas      264
Evolution, right      410
Exact contact structure      467
Exact differential forms      353
Expansion at x      311
Expansion property      311
Exponential law      445
Exponential mapping      372
Expose a subset      130
Extension of groups      412
Extension property      47
Extension property, scalar valued      221
Extension property, vector valued      221
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