1114:
6917:
885:
36:
4021:
2066:
You must be a novice in analysis or a genius like Nash to believe anything like that can be ever true. may strike you as realistic as a successful performance of perpetuum mobile with a mechanical implementation of
Maxwell's demon... unless you start following Nash's computation and realize to your
4597:
Presented directly as above, the meaning and naturality of the "tame" condition is rather obscure. The situation is clarified if one re-considers the basic examples given above, in which the relevant "exponentially decreasing" sequences in Banach spaces arise from restriction of a
Fourier transform.
5572:
1615:
borne out for the map which sends an immersion to its induced
Riemannian metric; given that this map is of order 1, one does not gain the "expected" one derivative upon inverting the operator. The same failure is common in geometric problems, where the action of the diffeomorphism group is the root
2291:
the latter of which reflects the forms given above. This is rather important, since the improved quadratic convergence of the "true" Newton iteration is significantly used to combat the error of "smoothing", in order to obtain convergence. Certain approaches, in particular Nash's and
Hamilton's,
2030:
transparently does not encounter the same difficulty as the previous "unsmoothed" version, since it is an iteration in the space of smooth functions which never loses regularity. So one has a well-defined sequence of functions; the major surprise of Nash's approach is that this sequence actually
1109:{\displaystyle {\widetilde {f}}\mapsto \sum _{\alpha =1}^{N}{\frac {\partial f^{\alpha }}{\partial u^{i}}}{\frac {\partial {\widetilde {f}}^{\beta }}{\partial u^{j}}}+\sum _{\alpha =1}^{N}{\frac {\partial {\widetilde {f}}^{\alpha }}{\partial u^{i}}}{\frac {\partial f^{\beta }}{\partial u^{j}}}.}
156:
In contrast to the Banach space case, in which the invertibility of the derivative at a point is sufficient for a map to be locally invertible, the Nash–Moser theorem requires the derivative to be invertible in a neighborhood. The theorem is widely used to prove local existence for non-linear
524:
4598:
Recall that smoothness of a function on
Euclidean space is directly related to the rate of decay of its Fourier transform. "Tameness" is thus seen as a condition which allows an abstraction of the idea of a "smoothing operator" on a function space. Given a Banach space
3874:
390:
1616:
cause, and in problems of hyperbolic differential equations, where even in the very simplest problems one does not have the naively expected smoothness of a solution. All of these difficulties provide common contexts for applications of the Nash–Moser theorem.
2074:
The true "smoothed Newton iteration" is a little more complicated than the above form, although there are a few inequivalent forms, depending on where one chooses to insert the smoothing operators. The primary difference is that one requires invertibility of
5868:
5356:
879:
3759:
5659:
3596:
395:
2028:
2058:. For many mathematicians, this is rather surprising, since the "fix" of throwing in a smoothing operator seems too superficial to overcome the deep problem in the standard Newton method. For instance, on this point
1825:
2699:
2289:
5664:
The fundamental example says that, on a compact smooth manifold, a nonlinear partial differential operator (possibly between sections of vector bundles over the manifold) is a smooth tame map; in this case,
2186:
276:
7035:
6953:
5281:
4970:
2292:
follow the solution of an ordinary differential equation in function space rather than an iteration in function space; the relation of the latter to the former is essentially that of the solution of
3228:
1303:
4889:
3111:
2622:
3812:
526:
In Nash's solution of the isometric embedding problem (as would be expected in the solutions of nonlinear partial differential equations) a major step is a statement of the schematic form "If
749:
4704:
5119:
4016:{\displaystyle \Sigma (B)={\Big \{}{\text{maps }}x:\mathbb {N} \to B{\text{ s.t. }}\sup _{k\in \mathbb {N} }e^{nk}\|x_{k}\|_{B}<\infty {\text{ for all }}n\in \mathbb {N} {\Big \}}.}
3049:
5751:
3630:
4448:
3004:
1611:; this causes no extra difficulty whatsoever for the application of the Banach space implicit function theorem. However, the above analysis shows that this naive expectation is
6034:
5221:
5074:
5048:
3347:
3306:
2971:
2382:
269:
6946:
5714:
3385:
2945:
2840:
2485:
4081:
4742:
4232:
2450:
6806:
4777:
2350:
1537:
1511:
75:
5930:
3486:
2796:
1544:
By exactly the same reasoning, one cannot directly apply the Banach space implicit function theorem even if one uses the Hölder spaces, the
Sobolev spaces, or any of the
4645:
3846:
2901:
691:
4307:
4199:
4117:
5185:
4972:
If one accepts the schematic idea of the proof devised by Nash, and in particular his use of smoothing operators, the "tame" condition then becomes rather reasonable.
1339:
4340:
4272:
3625:
2728:
1919:
239:
5969:
5903:
5350:
5020:
4497:
3454:
3137:
2757:
2519:
2408:
775:
5146:
4568:
4144:
3481:
649:
1146:
622:
573:
6939:
2866:
4668:
3869:
3160:
6469:
5324:
5304:
4832:
4812:
4616:
4588:
4541:
4521:
4471:
4404:
4380:
4360:
4252:
4164:
4045:
3428:
3408:
3265:
2571:
1668:
1593:
1423:
1375:
1186:
1166:
769:
593:
544:
5567:{\displaystyle {\big \|}D^{k}P\left(f,h_{1},\ldots ,h_{k}\right){\big \|}_{n}\leq C_{n}{\Big (}\|f\|_{n+r}+\|h_{1}\|_{n+r}+\cdots +\|h_{k}\|_{n+r}+1{\Big )}}
165:. It is particularly useful when the inverse to the derivative "loses" derivatives, and therefore the Banach space implicit function theorem cannot be used.
5578:
1729:
6632:
201:. However, it has proven quite difficult to find a suitable general formulation; there is, to date, no all-encompassing version; various versions due to
2627:
4450:
the space of smooth functions whose derivatives all vanish on the boundary, is a tamely graded Fréchet space, with any of the above graded structures.
7386:
6759:
6614:
7260:
7115:
6590:
2191:
2091:
1449:. The source of the problem can be quite succinctly phrased in the following way: the Gauss equation shows that there is a differential operator
696:
Following standard practice, one would expect to apply the Banach space inverse function theorem. So, for instance, one might expect to restrict
4894:
7250:
213:, Saint-Raymond, Schwartz, and Sergeraert are given in the references below. That of Hamilton's, quoted below, is particularly widely cited.
3165:
1191:
519:{\displaystyle P(f)_{ij}=\sum _{\alpha =1}^{N}{\frac {\partial f^{\alpha }}{\partial u^{i}}}{\frac {\partial f^{\alpha }}{\partial u^{j}}}.}
7007:
7040:
2305:
1895:, and so on. In finitely many steps the iteration must end, since it will lose all regularity and the next step will not even be defined.
1119:
However, there is a deep reason that such a formulation cannot work. The issue is that there is a second-order differential operator of
7376:
6482:
7074:
6571:
6462:
6081:
1116:
If one could show that this were invertible, with bounded inverse, then the Banach space inverse function theorem directly applies.
108:
6841:
5226:
7371:
7309:
7084:
6486:
1541:, cannot be bounded between appropriate Banach spaces, and hence the Banach space implicit function theorem cannot be applied.
198:
7304:
158:
6355:
Sergeraert, Francis (1972), "Un théorème de fonctions implicites sur certains espaces de Fréchet et quelques applications",
6637:
4837:
3054:
2578:
7245:
6693:
3764:
7161:
6920:
6642:
6627:
6455:
2529:
is locally injective. And if each linearization is only surjective, and a family of right inverses is smooth tame, then
385:{\displaystyle P:C^{1}(\Omega ;\mathbb {R} ^{N})\to C^{0}{\big (}\Omega ;{\text{Sym}}_{n\times n}(\mathbb {R} ){\big )}}
221:
This will be introduced in the original setting of the Nash–Moser theorem, that of the isometric embedding problem. Let
6657:
7153:
1703:; in the above language this reflects a "loss of one derivative". One can concretely see the failure of trying to use
1624:
This section only aims to describe an idea, and as such it is intentionally imprecise. For concreteness, suppose that
7228:
6902:
6662:
703:
4673:
7381:
7157:
6856:
6780:
6069:
6046:
2059:
202:
7212:
6897:
7299:
7141:
6713:
5079:
46:
6647:
7233:
7120:
6966:
6749:
6550:
3009:
141:
6622:
4499:
is a smooth vector bundle, then the space of smooth sections is tame, with any of the above graded structures.
5740:). (It is not too difficult to see that this is sufficient to prove the general case.) For a positive number
7335:
6846:
1385:) denotes its second fundamental form; the above equation is the Gauss equation from surface theory. So, if
7067:
6985:
6877:
6821:
6785:
4023:
The laboriousness of the definition is justified by the primary examples of tamely graded Fréchet spaces:
178:
4409:
7127:
7057:
6980:
6962:
2976:
122:
7045:
6419:(1976), "Generalized implicit function theorems with applications to some small divisor problems. II",
5994:
5198:
5053:
5025:
3311:
3270:
2950:
2525:
Similarly, if each linearization is only injective, and a family of left inverses is smooth tame, then
1599:. However, this is somewhat rare. In the case of uniformly elliptic differential operators, the famous
6388:(1975), "Generalized implicit function theorems with applications to some small divisor problems. I",
2355:
245:
7255:
7169:
7110:
7002:
6860:
6264:
5684:
3352:
2906:
2801:
2455:
4050:
7345:
7291:
7281:
7164:
7079:
6826:
6764:
6478:
6099:
4709:
4204:
2413:
2088:, and then one uses the "true" Newton iteration, corresponding to (using single-variable notation)
206:
194:
4747:
2329:
1520:
1494:
58:
7207:
7062:
6851:
6718:
6284:
6167:
5908:
2762:
1600:
5863:{\displaystyle f'=cS{\Big (}\theta _{t}(f),\theta _{t}{\big (}g_{\infty }-P(f){\big )}{\Big )}.}
4621:
3822:
2873:
1704:
654:
4590:
is defined by dyadic restriction of the
Fourier transform. The details are in pages 133-140 of
4279:
4171:
4089:
874:{\displaystyle C^{5}(\Omega ;\mathbb {R} ^{N})\to C^{4}(\Omega ;Sym_{n\times n}(\mathbb {R} ))}
189:), for instance, showed that Nash's methods could be successfully applied to solve problems on
6831:
6324:
6139:
6103:
6077:
5158:
3754:{\displaystyle \|M(\{x_{i}\})\|_{n}\leq C_{n}\sup _{k\in \mathbb {N} }e^{(r+n)k}\|x_{k}\|_{B}}
2293:
1308:
210:
4312:
4257:
3601:
2704:
224:
7350:
7197:
7187:
7136:
7089:
7050:
6836:
6754:
6723:
6703:
6688:
6683:
6678:
6515:
6428:
6397:
6364:
6336:
6276:
6187:
6151:
6118:
5939:
5873:
5329:
4999:
4476:
3433:
3116:
2736:
2494:
2387:
133:
6440:
6409:
6378:
6348:
6317:
6296:
6257:
6230:
6201:
6163:
6132:
6091:
6062:
5124:
4546:
4122:
3459:
627:
7325:
7202:
7192:
7017:
7012:
6698:
6652:
6600:
6595:
6566:
6447:
6436:
6405:
6374:
6344:
6313:
6292:
6253:
6226:
6197:
6159:
6128:
6087:
6058:
1122:
598:
549:
162:
148:
to settings when the required solution mapping for the linearized problem is not bounded.
6525:
6242:"A rapidly convergent iteration method and non-linear partial differential equations. II"
6237:
6210:
2845:
1603:
show that this naive expectation is borne out, with the caveat that one must replace the
137:
6931:
6241:
6215:"A rapidly convergent iteration method and non-linear partial differential equations. I"
6214:
4650:
3851:
3142:
1628:
is an order-one differential operator on some function spaces, so that it defines a map
7340:
7146:
6887:
6739:
6540:
6416:
6385:
5309:
5289:
4817:
4782:
4601:
4573:
4526:
4506:
4456:
4389:
4365:
4345:
4237:
4149:
4030:
3413:
3393:
3250:
2556:
1653:
1578:
1408:
1360:
1171:
1151:
754:
578:
529:
190:
4670:
the precise analogue of a smoothing operator can be defined in the following way. Let
7365:
7329:
7105:
6997:
6992:
6892:
6816:
6545:
6530:
6520:
6171:
4083:
is a tamely graded Fréchet space, when given any of the following graded structures:
130:
81:
6178:
Hörmander, L. (1977), "Correction to: "The boundary problems of physical geodesy"",
6123:
5654:{\displaystyle \left(f,h_{1},\dots ,h_{k}\right)\in U\times F\times \cdots \times F}
3591:{\displaystyle \sup _{k\in \mathbb {N} }e^{nk}\|L(f)_{k}\|_{B}\leq C_{n}\|f\|_{r+n}}
17:
7276:
7131:
6882:
6535:
6505:
145:
6076:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin.
6811:
6801:
6708:
6510:
2023:{\displaystyle f_{n+1}=f_{n}+S{\big (}\theta _{n}(g_{\infty }-P(f_{n})){\big )}}
6304:
Saint-Raymond, Xavier (1989), "A simple Nash-Moser implicit function theorem",
7027:
6744:
6584:
6580:
6576:
181:. It is clear from his paper that his method can be generalized. Moser (
4503:
To recognize the tame structure of these examples, one topologically embeds
6432:
6401:
6340:
1898:
Nash's solution is quite striking in its simplicity. Suppose that for each
5936:), then the solution of this differential equation with initial condition
7179:
6369:
5724:
are spaces of exponentially decreasing sequences in Banach spaces, i.e.
1912:
function, returns a smooth function, and approximates the identity when
1707:
to prove the Banach space implicit function theorem in this context: if
6288:
6192:
6155:
1548:
spaces. In any of these settings, an inverse to the linearization of
1148:
which coincides with a second-order differential operator applied to
6280:
1820:{\displaystyle f_{n+1}=f_{n}+S{\big (}g_{\infty }-P(f_{n}){\big )},}
1472:
In context, the upshot is that the inverse to the linearization of
3848:
denotes the vector space of exponentially decreasing sequences in
2694:{\displaystyle \|f\|_{0}\leq \|f\|_{1}\leq \|f\|_{2}\leq \cdots }
6935:
6451:
5428:
5362:
29:
6049:(1972), "Smoothing and inversion of differential operators",
4047:
is a compact smooth manifold (with or without boundary) then
2306:
Differentiation in Fréchet spaces § Tame Fréchet spaces
2284:{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{0})}},}
7036:
Differentiable vector–valued functions from
Euclidean space
5276:{\displaystyle D^{k}P:U\times F\times \cdots \times F\to G}
2181:{\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}}
6267:(1956), "The imbedding problem for Riemannian manifolds",
4965:{\displaystyle \left(\theta _{t}x\right)_{i}=s(t-i)x_{i}.}
575:
is positive-definite, then for any matrix-valued function
2730:
One requires these to satisfy the following conditions:
2533:
is locally surjective with a smooth tame right inverse.
90:
3223:{\displaystyle \lim _{j\to \infty }\|f_{j}-f\|_{n}=0.}
1298:{\displaystyle R^{P(f)}=|H(f)|^{2}-|h(f)|_{P(f)}^{2},}
6142:(1976), "The boundary problems of physical geodesy",
5997:
5942:
5911:
5876:
5754:
5687:
5581:
5359:
5332:
5312:
5292:
5229:
5201:
5161:
5127:
5082:
5056:
5028:
5002:
4897:
4840:
4820:
4785:
4750:
4712:
4676:
4653:
4624:
4604:
4576:
4549:
4529:
4509:
4479:
4459:
4412:
4392:
4368:
4348:
4315:
4282:
4260:
4240:
4207:
4174:
4152:
4125:
4092:
4053:
4033:
3877:
3854:
3825:
3767:
3633:
3604:
3489:
3462:
3436:
3416:
3396:
3355:
3314:
3273:
3253:
3168:
3145:
3119:
3057:
3012:
2979:
2953:
2909:
2876:
2848:
2804:
2765:
2739:
2707:
2630:
2581:
2559:
2497:
2458:
2416:
2390:
2358:
2332:
2194:
2094:
1922:
1732:
1656:
1581:
1523:
1497:
1411:
1363:
1311:
1194:
1174:
1154:
1125:
888:
778:
757:
706:
657:
630:
601:
581:
552:
532:
398:
279:
248:
227:
61:
4884:{\displaystyle \theta _{t}:\Sigma (B)\to \Sigma (B)}
3106:{\displaystyle \|f_{j}-f_{k}\|_{n}<\varepsilon ,}
2617:{\displaystyle \|\,\cdot \,\|_{n}:F\to \mathbb {R} }
2452:
is invertible, and the family of inverses, as a map
177:, who proved the theorem in the special case of the
7318:
7290:
7269:
7221:
7178:
7098:
7026:
6973:
6870:
6794:
6773:
6732:
6671:
6613:
6559:
6494:
5744:, consider the ordinary differential equation in Σ(
3807:{\displaystyle \left\{x_{i}\right\}\in \Sigma (B).}
6807:Spectral theory of ordinary differential equations
6028:
5963:
5924:
5897:
5862:
5708:
5653:
5566:
5344:
5318:
5298:
5275:
5215:
5179:
5140:
5113:
5068:
5042:
5014:
4964:
4883:
4826:
4806:
4771:
4736:
4698:
4662:
4639:
4610:
4582:
4562:
4535:
4515:
4491:
4465:
4442:
4398:
4374:
4354:
4334:
4301:
4266:
4246:
4226:
4193:
4158:
4138:
4111:
4075:
4039:
4015:
3863:
3840:
3806:
3753:
3619:
3590:
3475:
3448:
3422:
3402:
3379:
3341:
3300:
3259:
3222:
3154:
3131:
3105:
3043:
2998:
2965:
2939:
2895:
2860:
2834:
2790:
2751:
2722:
2693:
2616:
2565:
2513:
2479:
2444:
2402:
2376:
2344:
2283:
2180:
2022:
1819:
1662:
1587:
1559:. A very naive expectation is that, generally, if
1531:
1505:
1417:
1369:
1333:
1297:
1180:
1160:
1140:
1108:
873:
763:
743:
685:
643:
616:
587:
567:
538:
518:
384:
263:
233:
69:
6327:(1960), "On Nash's implicit functional theorem",
5852:
5774:
5559:
5454:
4005:
3895:
1341:is the scalar curvature of the Riemannian metric
6104:"The inverse function theorem of Nash and Moser"
4406:is a compact smooth manifold-with-boundary then
3928:
3686:
3491:
3170:
4570:functions on this Euclidean space, and the map
2316:
2064:
1916:is large. Then the "smoothed" Newton iteration
744:{\displaystyle C^{5}(\Omega ;\mathbb {R} ^{N})}
5669:can be taken to be the order of the operator.
4699:{\displaystyle s:\mathbb {R} \to \mathbb {R} }
2491:is locally invertible, and each local inverse
2084:for an entire open neighborhood of choices of
2067:immense surprise that the smoothing does work.
1357:) denotes the mean curvature of the immersion
51:In particular, it has problems with not using
27:Generalization of the inverse function theorem
6947:
6463:
6111:Bulletin of the American Mathematical Society
5845:
5813:
2015:
1960:
1809:
1770:
377:
335:
8:
5536:
5522:
5498:
5484:
5466:
5459:
5102:
5083:
4290:
4283:
4182:
4175:
4100:
4093:
3972:
3958:
3742:
3728:
3663:
3656:
3643:
3634:
3573:
3566:
3544:
3521:
3205:
3185:
3085:
3058:
2773:
2766:
2676:
2669:
2657:
2650:
2638:
2631:
2591:
2582:
2384:be a smooth tame map. Suppose that for each
5114:{\displaystyle \|f-f_{1}\|<\varepsilon }
771:in this domain, to study the linearization
6954:
6940:
6932:
6498:
6470:
6456:
6448:
2299:
1453:such that the order of the composition of
6368:
6191:
6122:
6017:
5996:
5941:
5916:
5910:
5875:
5851:
5850:
5844:
5843:
5822:
5812:
5811:
5805:
5783:
5773:
5772:
5753:
5686:
5616:
5597:
5580:
5558:
5557:
5539:
5529:
5501:
5491:
5469:
5453:
5452:
5446:
5433:
5427:
5426:
5414:
5395:
5371:
5361:
5360:
5358:
5331:
5311:
5291:
5234:
5228:
5209:
5208:
5200:
5160:
5132:
5126:
5096:
5081:
5055:
5036:
5035:
5027:
5001:
4953:
4922:
4908:
4896:
4845:
4839:
4819:
4784:
4749:
4711:
4692:
4691:
4684:
4683:
4675:
4652:
4647:of exponentially decreasing sequences in
4623:
4603:
4575:
4554:
4548:
4528:
4508:
4478:
4458:
4422:
4417:
4411:
4391:
4367:
4347:
4320:
4314:
4293:
4281:
4259:
4239:
4212:
4206:
4185:
4173:
4151:
4130:
4124:
4103:
4091:
4058:
4052:
4032:
4004:
4003:
3999:
3998:
3987:
3975:
3965:
3949:
3939:
3938:
3931:
3922:
3912:
3911:
3900:
3894:
3893:
3876:
3853:
3824:
3776:
3766:
3745:
3735:
3707:
3697:
3696:
3689:
3679:
3666:
3650:
3632:
3603:
3576:
3560:
3547:
3537:
3512:
3502:
3501:
3494:
3488:
3467:
3461:
3435:
3415:
3395:
3354:
3313:
3272:
3252:
3243:if it satisfies the following condition:
3208:
3192:
3173:
3167:
3144:
3118:
3088:
3078:
3065:
3056:
3044:{\displaystyle j,k>N_{n,\varepsilon }}
3029:
3011:
2984:
2978:
2952:
2908:
2881:
2875:
2847:
2803:
2776:
2764:
2738:
2706:
2679:
2660:
2641:
2629:
2610:
2609:
2594:
2589:
2585:
2580:
2558:
2502:
2496:
2457:
2424:
2415:
2389:
2357:
2331:
2266:
2240:
2227:
2218:
2199:
2193:
2166:
2140:
2127:
2118:
2099:
2093:
2014:
2013:
2001:
1982:
1969:
1959:
1958:
1946:
1927:
1921:
1808:
1807:
1798:
1779:
1769:
1768:
1756:
1737:
1731:
1655:
1580:
1525:
1524:
1522:
1499:
1498:
1496:
1410:
1405:. Then, according to the above equation,
1362:
1316:
1310:
1286:
1272:
1267:
1249:
1240:
1235:
1217:
1199:
1193:
1173:
1153:
1124:
1094:
1079:
1069:
1060:
1045:
1034:
1033:
1026:
1020:
1009:
993:
978:
967:
966:
959:
950:
935:
925:
919:
908:
890:
889:
887:
861:
860:
845:
820:
804:
800:
799:
783:
777:
756:
732:
728:
727:
711:
705:
668:
656:
635:
629:
600:
580:
551:
531:
504:
489:
479:
470:
455:
445:
439:
428:
412:
397:
376:
375:
368:
367:
352:
347:
334:
333:
327:
311:
307:
306:
290:
278:
255:
251:
250:
247:
226:
109:Learn how and when to remove this message
63:
62:
60:
7251:No infinite-dimensional Lebesgue measure
6760:Group algebra of a locally compact group
4591:
3235:Such a graded Fréchet space is called a
2311:
7261:Structure theorem for Gaussian measures
4706:be a smooth function which vanishes on
5681:denote the family of inverse mappings
4779:and takes values only in the interval
1461:is less than the sum of the orders of
186:
182:
173:The Nash–Moser theorem traces back to
7137:infinite-dimensional Gaussian measure
2300:Hamilton's formulation of the theorem
1620:The schematic form of Nash's solution
7:
7008:Infinite-dimensional vector function
2575:a countable collection of seminorms
2296:to that of a differential equation.
1902:>0 one has a smoothing operator θ
174:
4443:{\displaystyle C_{0}^{\infty }(M),}
3387:is the identity map and such that:
2310:The following statement appears in
6018:
5917:
5823:
4869:
4854:
4760:
4719:
4625:
4423:
4059:
3984:
3878:
3826:
3789:
3321:
3286:
3180:
2999:{\displaystyle N_{n,\varepsilon }}
2903:is a sequence such that, for each
1983:
1780:
1087:
1072:
1053:
1029:
986:
962:
943:
928:
829:
792:
720:
497:
482:
463:
448:
340:
299:
228:
217:The problem of loss of derivatives
25:
7075:Generalizations of the derivative
7041:Differentiation in Fréchet spaces
6029:{\displaystyle P(f)=g_{\infty }.}
5216:{\displaystyle k\in \mathbb {N} }
5069:{\displaystyle \varepsilon >0}
5043:{\displaystyle n\in \mathbb {N} }
4473:is a compact smooth manifold and
3342:{\displaystyle M:\Sigma (B)\to F}
3301:{\displaystyle L:F\to \Sigma (B)}
2966:{\displaystyle \varepsilon >0}
6916:
6915:
6842:Topological quantum field theory
2549:consists of the following data:
2377:{\displaystyle P:U\rightarrow G}
264:{\displaystyle \mathbb {R} ^{n}}
34:
7387:Theorems in functional analysis
7310:Holomorphic functional calculus
6246:Ann. Scuola Norm. Sup. Pisa (3)
6219:Ann. Scuola Norm. Sup. Pisa (3)
6124:10.1090/S0273-0979-1982-15004-2
5716:Consider the special case that
5709:{\displaystyle U\times G\to F.}
4744:is identically equal to one on
3380:{\displaystyle M\circ L:F\to F}
2940:{\displaystyle n=0,1,2,\ldots }
2835:{\displaystyle n=0,1,2,\ldots }
2480:{\displaystyle U\times G\to F,}
1567:differential operator, then if
7305:Continuous functional calculus
6357:Ann. Sci. École Norm. Sup. (4)
6074:Partial Differential Relations
6007:
6001:
5952:
5946:
5886:
5880:
5840:
5834:
5795:
5789:
5697:
5267:
5171:
4988:be graded Fréchet spaces. Let
4946:
4934:
4878:
4872:
4866:
4863:
4857:
4798:
4786:
4763:
4751:
4728:
4713:
4688:
4634:
4628:
4483:
4434:
4428:
4076:{\displaystyle C^{\infty }(M)}
4070:
4064:
3916:
3887:
3881:
3835:
3829:
3798:
3792:
3720:
3708:
3659:
3640:
3534:
3527:
3371:
3333:
3330:
3324:
3295:
3289:
3283:
3177:
2606:
2468:
2436:
2368:
2272:
2259:
2246:
2233:
2172:
2159:
2146:
2133:
2010:
2007:
1994:
1975:
1804:
1791:
1726:and one defines the iteration
1607:spaces with the Hölder spaces
1326:
1320:
1282:
1276:
1268:
1263:
1257:
1250:
1236:
1231:
1225:
1218:
1209:
1203:
1135:
1129:
901:
868:
865:
857:
826:
813:
810:
789:
738:
717:
674:
661:
611:
605:
562:
556:
409:
402:
372:
364:
320:
317:
296:
159:partial differential equations
43:This article needs editing to
1:
6638:Uniform boundedness principle
4737:{\displaystyle (-\infty ,0),}
4227:{\displaystyle C^{n,\alpha }}
2445:{\displaystyle dP_{f}:F\to G}
1476:, even if it exists as a map
140:, is a generalization of the
121:In the mathematical field of
4772:{\displaystyle (1,\infty ),}
4618:and the corresponding space
4543:is taken to be the space of
3247:there exists a Banach space
2345:{\displaystyle U\subseteq F}
2326:be tame Fréchet spaces, let
1532:{\displaystyle \mathbb {R} }
1506:{\displaystyle \mathbb {R} }
1445:| would have to be at least
77:consistently, and not using
70:{\displaystyle \mathbb {R} }
5932:is sufficiently small in Σ(
5925:{\displaystyle g_{\infty }}
2791:{\displaystyle \|f\|_{n}=0}
2352:be an open subset, and let
179:isometric embedding problem
7403:
6781:Invariant subspace problem
6180:Arch. Rational Mech. Anal.
6144:Arch. Rational Mech. Anal.
4814:Then for each real number
4640:{\displaystyle \Sigma (B)}
3841:{\displaystyle \Sigma (B)}
2896:{\displaystyle f_{j}\in F}
2303:
686:{\displaystyle P(f_{g})=g}
7377:Topological vector spaces
7300:Borel functional calculus
6967:topological vector spaces
6911:
6501:
5283:satisfies the following:
4302:{\displaystyle \|f\|_{n}}
4194:{\displaystyle \|f\|_{n}}
4112:{\displaystyle \|f\|_{n}}
1873:. By the same reasoning,
1552:will fail to be bounded.
7234:Inverse function theorem
7121:Classical Wiener measure
6750:Spectrum of a C*-algebra
5180:{\displaystyle P:U\to G}
4996:, meaning that for each
2031:converges to a function
1646:. Suppose that, at some
1619:
1334:{\displaystyle R^{P(f)}}
142:inverse function theorem
45:comply with Knowledge's
7336:Convenient vector space
6847:Noncommutative geometry
5870:Hamilton shows that if
4335:{\displaystyle W^{n,p}}
4267:{\displaystyle \alpha }
3620:{\displaystyle f\in F,}
2723:{\displaystyle f\in F.}
1555:This is the problem of
234:{\displaystyle \Omega }
7372:Differential equations
7229:Cameron–Martin theorem
6986:Classical Wiener space
6903:Tomita–Takesaki theory
6878:Approximation property
6822:Calculus of variations
6433:10.1002/cpa.3160290104
6421:Comm. Pure Appl. Math.
6402:10.1002/cpa.3160280104
6390:Comm. Pure Appl. Math.
6341:10.1002/cpa.3160130311
6329:Comm. Pure Appl. Math.
6030:
5965:
5964:{\displaystyle f(0)=0}
5926:
5899:
5898:{\displaystyle P(0)=0}
5864:
5710:
5655:
5568:
5346:
5345:{\displaystyle n>b}
5320:
5300:
5277:
5217:
5181:
5142:
5115:
5070:
5044:
5016:
5015:{\displaystyle f\in U}
4966:
4885:
4828:
4808:
4773:
4738:
4700:
4664:
4641:
4612:
4584:
4564:
4537:
4523:in a Euclidean space,
4517:
4493:
4492:{\displaystyle V\to M}
4467:
4444:
4400:
4376:
4356:
4336:
4303:
4268:
4248:
4228:
4195:
4160:
4140:
4113:
4077:
4041:
4017:
3865:
3842:
3808:
3755:
3621:
3592:
3477:
3450:
3449:{\displaystyle n>b}
3424:
3404:
3381:
3343:
3302:
3261:
3224:
3156:
3133:
3132:{\displaystyle f\in F}
3107:
3045:
3000:
2967:
2941:
2897:
2862:
2836:
2792:
2753:
2752:{\displaystyle f\in F}
2724:
2695:
2618:
2567:
2523:
2515:
2514:{\displaystyle P^{-1}}
2481:
2446:
2404:
2403:{\displaystyle f\in U}
2378:
2346:
2285:
2182:
2069:
2024:
1821:
1664:
1589:
1533:
1507:
1425:can generally be only
1419:
1371:
1335:
1299:
1182:
1162:
1142:
1110:
1025:
924:
875:
765:
751:and, for an immersion
745:
687:
645:
618:
589:
569:
540:
520:
444:
386:
265:
235:
136:and named for him and
71:
7246:Feldman–Hájek theorem
7058:Functional derivative
6981:Abstract Wiener space
6898:Banach–Mazur distance
6861:Generalized functions
6269:Annals of Mathematics
6031:
5966:
5927:
5900:
5865:
5711:
5656:
5569:
5347:
5321:
5301:
5278:
5218:
5182:
5148:is also contained in
5143:
5141:{\displaystyle f_{1}}
5116:
5071:
5045:
5017:
4992:be an open subset of
4967:
4886:
4829:
4809:
4774:
4739:
4701:
4665:
4642:
4613:
4585:
4565:
4563:{\displaystyle L^{1}}
4538:
4518:
4494:
4468:
4445:
4401:
4377:
4357:
4337:
4304:
4269:
4249:
4229:
4196:
4161:
4141:
4139:{\displaystyle C^{n}}
4114:
4078:
4042:
4018:
3866:
3843:
3809:
3756:
3622:
3593:
3478:
3476:{\displaystyle C_{n}}
3451:
3425:
3405:
3382:
3344:
3303:
3262:
3225:
3157:
3134:
3108:
3046:
3001:
2968:
2942:
2898:
2863:
2837:
2793:
2754:
2725:
2696:
2619:
2568:
2521:is a smooth tame map.
2516:
2487:is smooth tame. Then
2482:
2447:
2405:
2379:
2347:
2286:
2183:
2025:
1822:
1665:
1590:
1534:
1508:
1420:
1372:
1336:
1300:
1188:is an immersion then
1183:
1163:
1143:
1111:
1005:
904:
876:
766:
746:
688:
646:
644:{\displaystyle f_{g}}
619:
590:
570:
541:
521:
424:
387:
266:
241:be an open subset of
236:
87:for non-LaTeX markup.
72:
7170:Radonifying function
7111:Cylinder set measure
7003:Cylinder set measure
6643:Kakutani fixed-point
6628:Riesz representation
6100:Hamilton, Richard S.
5995:
5991:→∞ to a solution of
5971:exists as a mapping
5940:
5909:
5874:
5752:
5685:
5673:Proof of the theorem
5579:
5357:
5330:
5310:
5290:
5227:
5199:
5159:
5125:
5080:
5054:
5026:
5000:
4895:
4838:
4818:
4783:
4748:
4710:
4674:
4651:
4622:
4602:
4574:
4547:
4527:
4507:
4477:
4457:
4410:
4390:
4366:
4346:
4313:
4280:
4258:
4238:
4205:
4172:
4150:
4123:
4090:
4051:
4031:
3875:
3852:
3823:
3765:
3631:
3602:
3487:
3460:
3434:
3414:
3394:
3353:
3312:
3271:
3251:
3166:
3143:
3139:such that, for each
3117:
3055:
3010:
2977:
2951:
2907:
2874:
2846:
2802:
2763:
2737:
2705:
2628:
2579:
2557:
2545:graded Fréchet space
2495:
2456:
2414:
2388:
2356:
2330:
2192:
2092:
1920:
1730:
1689:has a right inverse
1670:, the linearization
1654:
1579:
1521:
1495:
1409:
1361:
1309:
1192:
1172:
1168:. To be precise: if
1152:
1141:{\displaystyle P(f)}
1123:
886:
776:
755:
704:
655:
628:
617:{\displaystyle P(f)}
599:
579:
568:{\displaystyle P(f)}
550:
530:
396:
277:
246:
225:
59:
18:Graded Fréchet space
7292:Functional calculus
7282:Covariance operator
7203:Gelfand–Pettis/Weak
7165:measurable function
7080:Hadamard derivative
6827:Functional calculus
6786:Mahler's conjecture
6765:Von Neumann algebra
6479:Functional analysis
6370:10.24033/asens.1239
4427:
3989: for all
3430:such that for each
2861:{\displaystyle f=0}
2537:Tame Fréchet spaces
1557:loss of derivatives
1291:
195:celestial mechanics
91:improve the content
7239:Nash–Moser theorem
7116:Canonical Gaussian
7063:Gateaux derivative
7046:Fréchet derivative
6852:Riemann hypothesis
6551:Topological vector
6306:Enseign. Math. (2)
6193:10.1007/BF00250435
6156:10.1007/BF00251855
6026:
5961:
5922:
5895:
5860:
5706:
5651:
5564:
5342:
5316:
5296:
5273:
5213:
5177:
5138:
5111:
5066:
5040:
5012:
4962:
4881:
4824:
4804:
4769:
4734:
4696:
4663:{\displaystyle B,}
4660:
4637:
4608:
4580:
4560:
4533:
4513:
4489:
4463:
4440:
4413:
4396:
4372:
4352:
4332:
4299:
4264:
4244:
4224:
4191:
4156:
4136:
4109:
4073:
4037:
4013:
3944:
3864:{\displaystyle B,}
3861:
3838:
3804:
3751:
3702:
3617:
3588:
3507:
3473:
3456:there is a number
3446:
3420:
3400:
3377:
3339:
3298:
3257:
3239:tame Fréchet space
3220:
3184:
3155:{\displaystyle n,}
3152:
3129:
3113:then there exists
3103:
3041:
2996:
2963:
2937:
2893:
2858:
2832:
2788:
2749:
2720:
2691:
2614:
2563:
2511:
2477:
2442:
2410:the linearization
2400:
2374:
2342:
2281:
2178:
2020:
1817:
1660:
1601:Schauder estimates
1585:
1529:
1503:
1415:
1401:is generally only
1367:
1331:
1295:
1266:
1178:
1158:
1138:
1106:
871:
761:
741:
683:
641:
614:
595:which is close to
585:
565:
536:
516:
382:
261:
231:
127:Nash–Moser theorem
67:
7382:Inverse functions
7359:
7358:
7256:Sazonov's theorem
7142:Projection-valued
6929:
6928:
6832:Integral operator
6609:
6608:
5319:{\displaystyle b}
5299:{\displaystyle r}
4827:{\displaystyle t}
4807:{\displaystyle .}
4611:{\displaystyle B}
4583:{\displaystyle L}
4536:{\displaystyle B}
4516:{\displaystyle M}
4466:{\displaystyle M}
4399:{\displaystyle M}
4375:{\displaystyle p}
4355:{\displaystyle f}
4247:{\displaystyle f}
4159:{\displaystyle f}
4040:{\displaystyle M}
3990:
3927:
3925:
3903:
3685:
3490:
3423:{\displaystyle b}
3403:{\displaystyle r}
3260:{\displaystyle B}
3169:
2566:{\displaystyle F}
2276:
2176:
1663:{\displaystyle f}
1588:{\displaystyle f}
1418:{\displaystyle f}
1370:{\displaystyle f}
1181:{\displaystyle f}
1161:{\displaystyle f}
1101:
1067:
1042:
1000:
975:
957:
898:
764:{\displaystyle f}
588:{\displaystyle g}
539:{\displaystyle f}
511:
477:
350:
273:Consider the map
119:
118:
111:
16:(Redirected from
7394:
7351:Hilbert manifold
7346:Fréchet manifold
7130: like
7090:Quasi-derivative
6956:
6949:
6942:
6933:
6919:
6918:
6837:Jones polynomial
6755:Operator algebra
6499:
6472:
6465:
6458:
6449:
6443:
6412:
6381:
6372:
6351:
6320:
6312:(3–4): 217–226,
6299:
6260:
6233:
6204:
6195:
6174:
6135:
6126:
6108:
6095:
6065:
6057:(130): 382–441,
6035:
6033:
6032:
6027:
6022:
6021:
5978:
5970:
5968:
5967:
5962:
5931:
5929:
5928:
5923:
5921:
5920:
5904:
5902:
5901:
5896:
5869:
5867:
5866:
5861:
5856:
5855:
5849:
5848:
5827:
5826:
5817:
5816:
5810:
5809:
5788:
5787:
5778:
5777:
5762:
5715:
5713:
5712:
5707:
5660:
5658:
5657:
5652:
5626:
5622:
5621:
5620:
5602:
5601:
5573:
5571:
5570:
5565:
5563:
5562:
5550:
5549:
5534:
5533:
5512:
5511:
5496:
5495:
5480:
5479:
5458:
5457:
5451:
5450:
5438:
5437:
5432:
5431:
5424:
5420:
5419:
5418:
5400:
5399:
5376:
5375:
5366:
5365:
5351:
5349:
5348:
5343:
5325:
5323:
5322:
5317:
5305:
5303:
5302:
5297:
5282:
5280:
5279:
5274:
5239:
5238:
5222:
5220:
5219:
5214:
5212:
5193:
5192:
5186:
5184:
5183:
5178:
5147:
5145:
5144:
5139:
5137:
5136:
5120:
5118:
5117:
5112:
5101:
5100:
5075:
5073:
5072:
5067:
5049:
5047:
5046:
5041:
5039:
5021:
5019:
5018:
5013:
4976:Smooth tame maps
4971:
4969:
4968:
4963:
4958:
4957:
4927:
4926:
4921:
4917:
4913:
4912:
4890:
4888:
4887:
4882:
4850:
4849:
4833:
4831:
4830:
4825:
4813:
4811:
4810:
4805:
4778:
4776:
4775:
4770:
4743:
4741:
4740:
4735:
4705:
4703:
4702:
4697:
4695:
4687:
4669:
4667:
4666:
4661:
4646:
4644:
4643:
4638:
4617:
4615:
4614:
4609:
4589:
4587:
4586:
4581:
4569:
4567:
4566:
4561:
4559:
4558:
4542:
4540:
4539:
4534:
4522:
4520:
4519:
4514:
4498:
4496:
4495:
4490:
4472:
4470:
4469:
4464:
4449:
4447:
4446:
4441:
4426:
4421:
4405:
4403:
4402:
4397:
4381:
4379:
4378:
4373:
4361:
4359:
4358:
4353:
4341:
4339:
4338:
4333:
4331:
4330:
4308:
4306:
4305:
4300:
4298:
4297:
4273:
4271:
4270:
4265:
4253:
4251:
4250:
4245:
4233:
4231:
4230:
4225:
4223:
4222:
4200:
4198:
4197:
4192:
4190:
4189:
4165:
4163:
4162:
4157:
4145:
4143:
4142:
4137:
4135:
4134:
4118:
4116:
4115:
4110:
4108:
4107:
4082:
4080:
4079:
4074:
4063:
4062:
4046:
4044:
4043:
4038:
4022:
4020:
4019:
4014:
4009:
4008:
4002:
3991:
3988:
3980:
3979:
3970:
3969:
3957:
3956:
3943:
3942:
3926:
3924: s.t.
3923:
3915:
3904:
3901:
3899:
3898:
3870:
3868:
3867:
3862:
3847:
3845:
3844:
3839:
3813:
3811:
3810:
3805:
3785:
3781:
3780:
3760:
3758:
3757:
3752:
3750:
3749:
3740:
3739:
3727:
3726:
3701:
3700:
3684:
3683:
3671:
3670:
3655:
3654:
3626:
3624:
3623:
3618:
3597:
3595:
3594:
3589:
3587:
3586:
3565:
3564:
3552:
3551:
3542:
3541:
3520:
3519:
3506:
3505:
3482:
3480:
3479:
3474:
3472:
3471:
3455:
3453:
3452:
3447:
3429:
3427:
3426:
3421:
3409:
3407:
3406:
3401:
3386:
3384:
3383:
3378:
3348:
3346:
3345:
3340:
3307:
3305:
3304:
3299:
3267:and linear maps
3266:
3264:
3263:
3258:
3241:
3240:
3229:
3227:
3226:
3221:
3213:
3212:
3197:
3196:
3183:
3161:
3159:
3158:
3153:
3138:
3136:
3135:
3130:
3112:
3110:
3109:
3104:
3093:
3092:
3083:
3082:
3070:
3069:
3050:
3048:
3047:
3042:
3040:
3039:
3005:
3003:
3002:
2997:
2995:
2994:
2972:
2970:
2969:
2964:
2946:
2944:
2943:
2938:
2902:
2900:
2899:
2894:
2886:
2885:
2867:
2865:
2864:
2859:
2841:
2839:
2838:
2833:
2797:
2795:
2794:
2789:
2781:
2780:
2758:
2756:
2755:
2750:
2729:
2727:
2726:
2721:
2700:
2698:
2697:
2692:
2684:
2683:
2665:
2664:
2646:
2645:
2623:
2621:
2620:
2615:
2613:
2599:
2598:
2572:
2570:
2569:
2564:
2547:
2546:
2520:
2518:
2517:
2512:
2510:
2509:
2486:
2484:
2483:
2478:
2451:
2449:
2448:
2443:
2429:
2428:
2409:
2407:
2406:
2401:
2383:
2381:
2380:
2375:
2351:
2349:
2348:
2343:
2290:
2288:
2287:
2282:
2277:
2275:
2271:
2270:
2258:
2249:
2245:
2244:
2228:
2223:
2222:
2210:
2209:
2187:
2185:
2184:
2179:
2177:
2175:
2171:
2170:
2158:
2149:
2145:
2144:
2128:
2123:
2122:
2110:
2109:
2057:
2029:
2027:
2026:
2021:
2019:
2018:
2006:
2005:
1987:
1986:
1974:
1973:
1964:
1963:
1951:
1950:
1938:
1937:
1826:
1824:
1823:
1818:
1813:
1812:
1803:
1802:
1784:
1783:
1774:
1773:
1761:
1760:
1748:
1747:
1702:
1688:
1669:
1667:
1666:
1661:
1641:
1598:
1594:
1592:
1591:
1586:
1574:
1570:
1562:
1551:
1547:
1540:
1538:
1536:
1535:
1530:
1528:
1512:
1510:
1509:
1504:
1502:
1475:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1440:
1436:
1432:
1424:
1422:
1421:
1416:
1376:
1374:
1373:
1368:
1340:
1338:
1337:
1332:
1330:
1329:
1304:
1302:
1301:
1296:
1290:
1285:
1271:
1253:
1245:
1244:
1239:
1221:
1213:
1212:
1187:
1185:
1184:
1179:
1167:
1165:
1164:
1159:
1147:
1145:
1144:
1139:
1115:
1113:
1112:
1107:
1102:
1100:
1099:
1098:
1085:
1084:
1083:
1070:
1068:
1066:
1065:
1064:
1051:
1050:
1049:
1044:
1043:
1035:
1027:
1024:
1019:
1001:
999:
998:
997:
984:
983:
982:
977:
976:
968:
960:
958:
956:
955:
954:
941:
940:
939:
926:
923:
918:
900:
899:
891:
881:
880:
878:
877:
872:
864:
856:
855:
825:
824:
809:
808:
803:
788:
787:
770:
768:
767:
762:
750:
748:
747:
742:
737:
736:
731:
716:
715:
699:
692:
690:
689:
684:
673:
672:
650:
648:
647:
642:
640:
639:
623:
621:
620:
615:
594:
592:
591:
586:
574:
572:
571:
566:
545:
543:
542:
537:
525:
523:
522:
517:
512:
510:
509:
508:
495:
494:
493:
480:
478:
476:
475:
474:
461:
460:
459:
446:
443:
438:
420:
419:
391:
389:
388:
383:
381:
380:
371:
363:
362:
351:
348:
339:
338:
332:
331:
316:
315:
310:
295:
294:
272:
270:
268:
267:
262:
260:
259:
254:
240:
238:
237:
232:
163:smooth functions
134:John Forbes Nash
129:, discovered by
114:
107:
103:
100:
94:
86:
80:
76:
74:
73:
68:
66:
38:
37:
30:
21:
7402:
7401:
7397:
7396:
7395:
7393:
7392:
7391:
7362:
7361:
7360:
7355:
7326:Banach manifold
7314:
7286:
7265:
7217:
7193:Direct integral
7174:
7094:
7022:
7018:Vector calculus
7013:Matrix calculus
6969:
6960:
6930:
6925:
6907:
6871:Advanced topics
6866:
6790:
6769:
6728:
6694:Hilbert–Schmidt
6667:
6658:Gelfand–Naimark
6605:
6555:
6490:
6476:
6446:
6415:
6384:
6354:
6323:
6303:
6281:10.2307/1969989
6263:
6236:
6209:
6177:
6140:Hörmander, Lars
6138:
6106:
6098:
6084:
6070:Gromov, Mikhael
6068:
6045:
6041:
6013:
5993:
5992:
5987:) converges as
5972:
5938:
5937:
5912:
5907:
5906:
5872:
5871:
5818:
5801:
5779:
5755:
5750:
5749:
5683:
5682:
5675:
5662:
5612:
5593:
5586:
5582:
5577:
5576:
5535:
5525:
5497:
5487:
5465:
5442:
5425:
5410:
5391:
5384:
5380:
5367:
5355:
5354:
5328:
5327:
5308:
5307:
5288:
5287:
5230:
5225:
5224:
5223:the derivative
5197:
5196:
5191:tame smooth map
5190:
5189:
5157:
5156:
5128:
5123:
5122:
5092:
5078:
5077:
5052:
5051:
5024:
5023:
4998:
4997:
4978:
4949:
4904:
4903:
4899:
4898:
4893:
4892:
4841:
4836:
4835:
4816:
4815:
4781:
4780:
4746:
4745:
4708:
4707:
4672:
4671:
4649:
4648:
4620:
4619:
4600:
4599:
4592:Hamilton (1982)
4572:
4571:
4550:
4545:
4544:
4525:
4524:
4505:
4504:
4475:
4474:
4455:
4454:
4408:
4407:
4388:
4387:
4364:
4363:
4344:
4343:
4316:
4311:
4310:
4289:
4278:
4277:
4256:
4255:
4236:
4235:
4208:
4203:
4202:
4181:
4170:
4169:
4148:
4147:
4126:
4121:
4120:
4099:
4088:
4087:
4054:
4049:
4048:
4029:
4028:
3971:
3961:
3945:
3873:
3872:
3850:
3849:
3821:
3820:
3772:
3768:
3763:
3762:
3741:
3731:
3703:
3675:
3662:
3646:
3629:
3628:
3600:
3599:
3572:
3556:
3543:
3533:
3508:
3485:
3484:
3463:
3458:
3457:
3432:
3431:
3412:
3411:
3392:
3391:
3351:
3350:
3310:
3309:
3269:
3268:
3249:
3248:
3238:
3237:
3204:
3188:
3164:
3163:
3141:
3140:
3115:
3114:
3084:
3074:
3061:
3053:
3052:
3025:
3008:
3007:
2980:
2975:
2974:
2949:
2948:
2905:
2904:
2877:
2872:
2871:
2844:
2843:
2800:
2799:
2772:
2761:
2760:
2735:
2734:
2703:
2702:
2675:
2656:
2637:
2626:
2625:
2590:
2577:
2576:
2555:
2554:
2553:a vector space
2544:
2543:
2539:
2498:
2493:
2492:
2454:
2453:
2420:
2412:
2411:
2386:
2385:
2354:
2353:
2328:
2327:
2312:Hamilton (1982)
2308:
2302:
2262:
2251:
2250:
2236:
2229:
2214:
2195:
2190:
2189:
2162:
2151:
2150:
2136:
2129:
2114:
2095:
2090:
2089:
2083:
2056:
2049:
2039:
2037:
1997:
1978:
1965:
1942:
1923:
1918:
1917:
1907:
1890:
1879:
1868:
1857:
1844:
1833:
1794:
1775:
1752:
1733:
1728:
1727:
1713:
1705:Newton's method
1690:
1679:
1671:
1652:
1651:
1629:
1622:
1596:
1577:
1576:
1572:
1568:
1560:
1549:
1545:
1519:
1518:
1493:
1492:
1490:
1477:
1473:
1466:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1434:
1430:
1407:
1406:
1359:
1358:
1312:
1307:
1306:
1234:
1195:
1190:
1189:
1170:
1169:
1150:
1149:
1121:
1120:
1090:
1086:
1075:
1071:
1056:
1052:
1032:
1028:
989:
985:
965:
961:
946:
942:
931:
927:
884:
883:
841:
816:
798:
779:
774:
773:
772:
753:
752:
726:
707:
702:
701:
697:
664:
653:
652:
631:
626:
625:
624:, there exists
597:
596:
577:
576:
548:
547:
528:
527:
500:
496:
485:
481:
466:
462:
451:
447:
408:
394:
393:
346:
323:
305:
286:
275:
274:
249:
244:
243:
242:
223:
222:
219:
191:periodic orbits
171:
154:
115:
104:
98:
95:
88:
84:
78:
57:
56:
47:Manual of Style
39:
35:
28:
23:
22:
15:
12:
11:
5:
7400:
7398:
7390:
7389:
7384:
7379:
7374:
7364:
7363:
7357:
7356:
7354:
7353:
7348:
7343:
7341:Choquet theory
7338:
7333:
7322:
7320:
7316:
7315:
7313:
7312:
7307:
7302:
7296:
7294:
7288:
7287:
7285:
7284:
7279:
7273:
7271:
7267:
7266:
7264:
7263:
7258:
7253:
7248:
7243:
7242:
7241:
7231:
7225:
7223:
7219:
7218:
7216:
7215:
7210:
7205:
7200:
7195:
7190:
7184:
7182:
7176:
7175:
7173:
7172:
7167:
7151:
7150:
7149:
7144:
7139:
7125:
7124:
7123:
7118:
7108:
7102:
7100:
7096:
7095:
7093:
7092:
7087:
7082:
7077:
7072:
7071:
7070:
7060:
7055:
7054:
7053:
7043:
7038:
7032:
7030:
7024:
7023:
7021:
7020:
7015:
7010:
7005:
7000:
6995:
6990:
6989:
6988:
6977:
6975:
6974:Basic concepts
6971:
6970:
6961:
6959:
6958:
6951:
6944:
6936:
6927:
6926:
6924:
6923:
6912:
6909:
6908:
6906:
6905:
6900:
6895:
6890:
6888:Choquet theory
6885:
6880:
6874:
6872:
6868:
6867:
6865:
6864:
6854:
6849:
6844:
6839:
6834:
6829:
6824:
6819:
6814:
6809:
6804:
6798:
6796:
6792:
6791:
6789:
6788:
6783:
6777:
6775:
6771:
6770:
6768:
6767:
6762:
6757:
6752:
6747:
6742:
6740:Banach algebra
6736:
6734:
6730:
6729:
6727:
6726:
6721:
6716:
6711:
6706:
6701:
6696:
6691:
6686:
6681:
6675:
6673:
6669:
6668:
6666:
6665:
6663:Banach–Alaoglu
6660:
6655:
6650:
6645:
6640:
6635:
6630:
6625:
6619:
6617:
6611:
6610:
6607:
6606:
6604:
6603:
6598:
6593:
6591:Locally convex
6588:
6574:
6569:
6563:
6561:
6557:
6556:
6554:
6553:
6548:
6543:
6538:
6533:
6528:
6523:
6518:
6513:
6508:
6502:
6496:
6492:
6491:
6477:
6475:
6474:
6467:
6460:
6452:
6445:
6444:
6413:
6382:
6363:(4): 599–660,
6352:
6335:(3): 509–530,
6321:
6301:
6261:
6234:
6207:
6206:
6205:
6136:
6113:, New Series,
6096:
6082:
6066:
6053:, New Series,
6042:
6040:
6037:
6025:
6020:
6016:
6012:
6009:
6006:
6003:
6000:
5960:
5957:
5954:
5951:
5948:
5945:
5919:
5915:
5894:
5891:
5888:
5885:
5882:
5879:
5859:
5854:
5847:
5842:
5839:
5836:
5833:
5830:
5825:
5821:
5815:
5808:
5804:
5800:
5797:
5794:
5791:
5786:
5782:
5776:
5771:
5768:
5765:
5761:
5758:
5705:
5702:
5699:
5696:
5693:
5690:
5674:
5671:
5650:
5647:
5644:
5641:
5638:
5635:
5632:
5629:
5625:
5619:
5615:
5611:
5608:
5605:
5600:
5596:
5592:
5589:
5585:
5561:
5556:
5553:
5548:
5545:
5542:
5538:
5532:
5528:
5524:
5521:
5518:
5515:
5510:
5507:
5504:
5500:
5494:
5490:
5486:
5483:
5478:
5475:
5472:
5468:
5464:
5461:
5456:
5449:
5445:
5441:
5436:
5430:
5423:
5417:
5413:
5409:
5406:
5403:
5398:
5394:
5390:
5387:
5383:
5379:
5374:
5370:
5364:
5341:
5338:
5335:
5315:
5295:
5285:
5272:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5248:
5245:
5242:
5237:
5233:
5211:
5207:
5204:
5176:
5173:
5170:
5167:
5164:
5135:
5131:
5110:
5107:
5104:
5099:
5095:
5091:
5088:
5085:
5065:
5062:
5059:
5038:
5034:
5031:
5011:
5008:
5005:
4977:
4974:
4961:
4956:
4952:
4948:
4945:
4942:
4939:
4936:
4933:
4930:
4925:
4920:
4916:
4911:
4907:
4902:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4848:
4844:
4823:
4803:
4800:
4797:
4794:
4791:
4788:
4768:
4765:
4762:
4759:
4756:
4753:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4694:
4690:
4686:
4682:
4679:
4659:
4656:
4636:
4633:
4630:
4627:
4607:
4579:
4557:
4553:
4532:
4512:
4501:
4500:
4488:
4485:
4482:
4462:
4451:
4439:
4436:
4433:
4430:
4425:
4420:
4416:
4395:
4384:
4383:
4382:
4371:
4351:
4329:
4326:
4323:
4319:
4296:
4292:
4288:
4285:
4274:
4263:
4243:
4221:
4218:
4215:
4211:
4188:
4184:
4180:
4177:
4166:
4155:
4133:
4129:
4106:
4102:
4098:
4095:
4072:
4069:
4066:
4061:
4057:
4036:
4012:
4007:
4001:
3997:
3994:
3986:
3983:
3978:
3974:
3968:
3964:
3960:
3955:
3952:
3948:
3941:
3937:
3934:
3930:
3921:
3918:
3914:
3910:
3907:
3897:
3892:
3889:
3886:
3883:
3880:
3860:
3857:
3837:
3834:
3831:
3828:
3817:
3816:
3815:
3814:
3803:
3800:
3797:
3794:
3791:
3788:
3784:
3779:
3775:
3771:
3748:
3744:
3738:
3734:
3730:
3725:
3722:
3719:
3716:
3713:
3710:
3706:
3699:
3695:
3692:
3688:
3682:
3678:
3674:
3669:
3665:
3661:
3658:
3653:
3649:
3645:
3642:
3639:
3636:
3616:
3613:
3610:
3607:
3585:
3582:
3579:
3575:
3571:
3568:
3563:
3559:
3555:
3550:
3546:
3540:
3536:
3532:
3529:
3526:
3523:
3518:
3515:
3511:
3504:
3500:
3497:
3493:
3470:
3466:
3445:
3442:
3439:
3419:
3399:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3256:
3233:
3232:
3231:
3230:
3219:
3216:
3211:
3207:
3203:
3200:
3195:
3191:
3187:
3182:
3179:
3176:
3172:
3151:
3148:
3128:
3125:
3122:
3102:
3099:
3096:
3091:
3087:
3081:
3077:
3073:
3068:
3064:
3060:
3038:
3035:
3032:
3028:
3024:
3021:
3018:
3015:
2993:
2990:
2987:
2983:
2962:
2959:
2956:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2892:
2889:
2884:
2880:
2868:
2857:
2854:
2851:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2787:
2784:
2779:
2775:
2771:
2768:
2748:
2745:
2742:
2719:
2716:
2713:
2710:
2690:
2687:
2682:
2678:
2674:
2671:
2668:
2663:
2659:
2655:
2652:
2649:
2644:
2640:
2636:
2633:
2612:
2608:
2605:
2602:
2597:
2593:
2588:
2584:
2573:
2562:
2538:
2535:
2508:
2505:
2501:
2476:
2473:
2470:
2467:
2464:
2461:
2441:
2438:
2435:
2432:
2427:
2423:
2419:
2399:
2396:
2393:
2373:
2370:
2367:
2364:
2361:
2341:
2338:
2335:
2301:
2298:
2294:Euler's method
2280:
2274:
2269:
2265:
2261:
2257:
2254:
2248:
2243:
2239:
2235:
2232:
2226:
2221:
2217:
2213:
2208:
2205:
2202:
2198:
2188:as opposed to
2174:
2169:
2165:
2161:
2157:
2154:
2148:
2143:
2139:
2135:
2132:
2126:
2121:
2117:
2113:
2108:
2105:
2102:
2098:
2079:
2060:Mikhael Gromov
2054:
2047:
2035:
2017:
2012:
2009:
2004:
2000:
1996:
1993:
1990:
1985:
1981:
1977:
1972:
1968:
1962:
1957:
1954:
1949:
1945:
1941:
1936:
1933:
1930:
1926:
1908:which takes a
1903:
1888:
1877:
1866:
1853:
1842:
1831:
1816:
1811:
1806:
1801:
1797:
1793:
1790:
1787:
1782:
1778:
1772:
1767:
1764:
1759:
1755:
1751:
1746:
1743:
1740:
1736:
1711:
1675:
1659:
1621:
1618:
1584:
1527:
1501:
1482:
1414:
1366:
1328:
1325:
1322:
1319:
1315:
1294:
1289:
1284:
1281:
1278:
1275:
1270:
1265:
1262:
1259:
1256:
1252:
1248:
1243:
1238:
1233:
1230:
1227:
1224:
1220:
1216:
1211:
1208:
1205:
1202:
1198:
1177:
1157:
1137:
1134:
1131:
1128:
1105:
1097:
1093:
1089:
1082:
1078:
1074:
1063:
1059:
1055:
1048:
1041:
1038:
1031:
1023:
1018:
1015:
1012:
1008:
1004:
996:
992:
988:
981:
974:
971:
964:
953:
949:
945:
938:
934:
930:
922:
917:
914:
911:
907:
903:
897:
894:
870:
867:
863:
859:
854:
851:
848:
844:
840:
837:
834:
831:
828:
823:
819:
815:
812:
807:
802:
797:
794:
791:
786:
782:
760:
740:
735:
730:
725:
722:
719:
714:
710:
682:
679:
676:
671:
667:
663:
660:
638:
634:
613:
610:
607:
604:
584:
564:
561:
558:
555:
535:
515:
507:
503:
499:
492:
488:
484:
473:
469:
465:
458:
454:
450:
442:
437:
434:
431:
427:
423:
418:
415:
411:
407:
404:
401:
379:
374:
370:
366:
361:
358:
355:
345:
342:
337:
330:
326:
322:
319:
314:
309:
304:
301:
298:
293:
289:
285:
282:
258:
253:
230:
218:
215:
170:
167:
153:
150:
117:
116:
65:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7399:
7388:
7385:
7383:
7380:
7378:
7375:
7373:
7370:
7369:
7367:
7352:
7349:
7347:
7344:
7342:
7339:
7337:
7334:
7331:
7327:
7324:
7323:
7321:
7317:
7311:
7308:
7306:
7303:
7301:
7298:
7297:
7295:
7293:
7289:
7283:
7280:
7278:
7275:
7274:
7272:
7268:
7262:
7259:
7257:
7254:
7252:
7249:
7247:
7244:
7240:
7237:
7236:
7235:
7232:
7230:
7227:
7226:
7224:
7220:
7214:
7211:
7209:
7206:
7204:
7201:
7199:
7196:
7194:
7191:
7189:
7186:
7185:
7183:
7181:
7177:
7171:
7168:
7166:
7163:
7159:
7155:
7152:
7148:
7145:
7143:
7140:
7138:
7135:
7134:
7133:
7132:set functions
7129:
7126:
7122:
7119:
7117:
7114:
7113:
7112:
7109:
7107:
7106:Besov measure
7104:
7103:
7101:
7099:Measurability
7097:
7091:
7088:
7086:
7083:
7081:
7078:
7076:
7073:
7069:
7066:
7065:
7064:
7061:
7059:
7056:
7052:
7049:
7048:
7047:
7044:
7042:
7039:
7037:
7034:
7033:
7031:
7029:
7025:
7019:
7016:
7014:
7011:
7009:
7006:
7004:
7001:
6999:
6998:Convex series
6996:
6994:
6993:Bochner space
6991:
6987:
6984:
6983:
6982:
6979:
6978:
6976:
6972:
6968:
6964:
6957:
6952:
6950:
6945:
6943:
6938:
6937:
6934:
6922:
6914:
6913:
6910:
6904:
6901:
6899:
6896:
6894:
6893:Weak topology
6891:
6889:
6886:
6884:
6881:
6879:
6876:
6875:
6873:
6869:
6862:
6858:
6855:
6853:
6850:
6848:
6845:
6843:
6840:
6838:
6835:
6833:
6830:
6828:
6825:
6823:
6820:
6818:
6817:Index theorem
6815:
6813:
6810:
6808:
6805:
6803:
6800:
6799:
6797:
6793:
6787:
6784:
6782:
6779:
6778:
6776:
6774:Open problems
6772:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6737:
6735:
6731:
6725:
6722:
6720:
6717:
6715:
6712:
6710:
6707:
6705:
6702:
6700:
6697:
6695:
6692:
6690:
6687:
6685:
6682:
6680:
6677:
6676:
6674:
6670:
6664:
6661:
6659:
6656:
6654:
6651:
6649:
6646:
6644:
6641:
6639:
6636:
6634:
6631:
6629:
6626:
6624:
6621:
6620:
6618:
6616:
6612:
6602:
6599:
6597:
6594:
6592:
6589:
6586:
6582:
6578:
6575:
6573:
6570:
6568:
6565:
6564:
6562:
6558:
6552:
6549:
6547:
6544:
6542:
6539:
6537:
6534:
6532:
6529:
6527:
6524:
6522:
6519:
6517:
6514:
6512:
6509:
6507:
6504:
6503:
6500:
6497:
6493:
6488:
6484:
6480:
6473:
6468:
6466:
6461:
6459:
6454:
6453:
6450:
6442:
6438:
6434:
6430:
6427:(1): 49–111,
6426:
6422:
6418:
6414:
6411:
6407:
6403:
6399:
6395:
6391:
6387:
6383:
6380:
6376:
6371:
6366:
6362:
6358:
6353:
6350:
6346:
6342:
6338:
6334:
6330:
6326:
6322:
6319:
6315:
6311:
6307:
6302:
6298:
6294:
6290:
6286:
6282:
6278:
6274:
6270:
6266:
6262:
6259:
6255:
6251:
6247:
6243:
6239:
6238:Moser, Jürgen
6235:
6232:
6228:
6224:
6220:
6216:
6212:
6211:Moser, Jürgen
6208:
6203:
6199:
6194:
6189:
6185:
6181:
6176:
6175:
6173:
6169:
6165:
6161:
6157:
6153:
6149:
6145:
6141:
6137:
6134:
6130:
6125:
6120:
6117:(1): 65–222,
6116:
6112:
6105:
6101:
6097:
6093:
6089:
6085:
6083:3-540-12177-3
6079:
6075:
6071:
6067:
6064:
6060:
6056:
6052:
6048:
6047:Gromov, M. L.
6044:
6043:
6038:
6036:
6023:
6014:
6010:
6004:
5998:
5990:
5986:
5982:
5976:
5958:
5955:
5949:
5943:
5935:
5913:
5892:
5889:
5883:
5877:
5857:
5837:
5831:
5828:
5819:
5806:
5802:
5798:
5792:
5784:
5780:
5769:
5766:
5763:
5759:
5756:
5747:
5743:
5739:
5735:
5731:
5727:
5723:
5719:
5703:
5700:
5694:
5691:
5688:
5680:
5672:
5670:
5668:
5648:
5645:
5642:
5639:
5636:
5633:
5630:
5627:
5623:
5617:
5613:
5609:
5606:
5603:
5598:
5594:
5590:
5587:
5583:
5574:
5554:
5551:
5546:
5543:
5540:
5530:
5526:
5519:
5516:
5513:
5508:
5505:
5502:
5492:
5488:
5481:
5476:
5473:
5470:
5462:
5447:
5443:
5439:
5434:
5421:
5415:
5411:
5407:
5404:
5401:
5396:
5392:
5388:
5385:
5381:
5377:
5372:
5368:
5339:
5336:
5333:
5313:
5293:
5284:
5270:
5264:
5261:
5258:
5255:
5252:
5249:
5246:
5243:
5240:
5235:
5231:
5205:
5202:
5194:
5174:
5168:
5165:
5162:
5155:A smooth map
5153:
5151:
5133:
5129:
5121:implies that
5108:
5105:
5097:
5093:
5089:
5086:
5063:
5060:
5057:
5032:
5029:
5009:
5006:
5003:
4995:
4991:
4987:
4983:
4975:
4973:
4959:
4954:
4950:
4943:
4940:
4937:
4931:
4928:
4923:
4918:
4914:
4909:
4905:
4900:
4875:
4860:
4851:
4846:
4842:
4821:
4801:
4795:
4792:
4789:
4766:
4757:
4754:
4731:
4725:
4722:
4716:
4680:
4677:
4657:
4654:
4631:
4605:
4595:
4593:
4577:
4555:
4551:
4530:
4510:
4486:
4480:
4460:
4452:
4437:
4431:
4418:
4414:
4393:
4385:
4369:
4349:
4327:
4324:
4321:
4317:
4294:
4286:
4275:
4261:
4241:
4219:
4216:
4213:
4209:
4186:
4178:
4167:
4153:
4131:
4127:
4104:
4096:
4085:
4084:
4067:
4055:
4034:
4026:
4025:
4024:
4010:
3995:
3992:
3981:
3976:
3966:
3962:
3953:
3950:
3946:
3935:
3932:
3919:
3908:
3905:
3890:
3884:
3858:
3855:
3832:
3801:
3795:
3786:
3782:
3777:
3773:
3769:
3746:
3736:
3732:
3723:
3717:
3714:
3711:
3704:
3693:
3690:
3680:
3676:
3672:
3667:
3651:
3647:
3637:
3614:
3611:
3608:
3605:
3583:
3580:
3577:
3569:
3561:
3557:
3553:
3548:
3538:
3530:
3524:
3516:
3513:
3509:
3498:
3495:
3468:
3464:
3443:
3440:
3437:
3417:
3397:
3390:there exists
3389:
3388:
3374:
3368:
3365:
3362:
3359:
3356:
3336:
3327:
3318:
3315:
3292:
3280:
3277:
3274:
3254:
3246:
3245:
3244:
3242:
3217:
3214:
3209:
3201:
3198:
3193:
3189:
3174:
3149:
3146:
3126:
3123:
3120:
3100:
3097:
3094:
3089:
3079:
3075:
3071:
3066:
3062:
3036:
3033:
3030:
3026:
3022:
3019:
3016:
3013:
2991:
2988:
2985:
2981:
2973:there exists
2960:
2957:
2954:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2890:
2887:
2882:
2878:
2869:
2855:
2852:
2849:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2805:
2785:
2782:
2777:
2769:
2759:is such that
2746:
2743:
2740:
2732:
2731:
2717:
2714:
2711:
2708:
2688:
2685:
2680:
2672:
2666:
2661:
2653:
2647:
2642:
2634:
2603:
2600:
2595:
2586:
2574:
2560:
2552:
2551:
2550:
2548:
2536:
2534:
2532:
2528:
2522:
2506:
2503:
2499:
2490:
2474:
2471:
2465:
2462:
2459:
2439:
2433:
2430:
2425:
2421:
2417:
2397:
2394:
2391:
2371:
2365:
2362:
2359:
2339:
2336:
2333:
2325:
2321:
2315:
2313:
2307:
2297:
2295:
2278:
2267:
2263:
2255:
2252:
2241:
2237:
2230:
2224:
2219:
2215:
2211:
2206:
2203:
2200:
2196:
2167:
2163:
2155:
2152:
2141:
2137:
2130:
2124:
2119:
2115:
2111:
2106:
2103:
2100:
2096:
2087:
2082:
2078:
2073:
2068:
2063:
2061:
2053:
2046:
2042:
2034:
2002:
1998:
1991:
1988:
1979:
1970:
1966:
1955:
1952:
1947:
1943:
1939:
1934:
1931:
1928:
1924:
1915:
1911:
1906:
1901:
1896:
1894:
1887:
1883:
1876:
1872:
1865:
1861:
1856:
1852:
1848:
1841:
1838:implies that
1837:
1830:
1814:
1799:
1795:
1788:
1785:
1776:
1765:
1762:
1757:
1753:
1749:
1744:
1741:
1738:
1734:
1725:
1721:
1717:
1710:
1706:
1701:
1697:
1693:
1687:
1683:
1678:
1674:
1657:
1649:
1645:
1640:
1636:
1632:
1627:
1617:
1614:
1610:
1606:
1602:
1582:
1566:
1558:
1553:
1542:
1516:
1489:
1485:
1480:
1470:
1429:; if it were
1428:
1412:
1404:
1400:
1396:
1392:
1388:
1384:
1380:
1364:
1356:
1352:
1348:
1344:
1323:
1317:
1313:
1292:
1287:
1279:
1273:
1260:
1254:
1246:
1241:
1228:
1222:
1214:
1206:
1200:
1196:
1175:
1155:
1132:
1126:
1117:
1103:
1095:
1091:
1080:
1076:
1061:
1057:
1046:
1039:
1036:
1021:
1016:
1013:
1010:
1006:
1002:
994:
990:
979:
972:
969:
951:
947:
936:
932:
920:
915:
912:
909:
905:
895:
892:
852:
849:
846:
842:
838:
835:
832:
821:
817:
805:
795:
784:
780:
758:
733:
723:
712:
708:
694:
680:
677:
669:
665:
658:
636:
632:
608:
602:
582:
559:
553:
546:is such that
533:
513:
505:
501:
490:
486:
471:
467:
456:
452:
440:
435:
432:
429:
425:
421:
416:
413:
405:
399:
359:
356:
353:
343:
328:
324:
312:
302:
291:
287:
283:
280:
256:
216:
214:
212:
208:
204:
200:
196:
192:
188:
184:
180:
176:
168:
166:
164:
161:in spaces of
160:
151:
149:
147:
146:Banach spaces
143:
139:
135:
132:
131:mathematician
128:
124:
113:
110:
102:
92:
83:
54:
50:
48:
41:
32:
31:
19:
7319:Applications
7277:Crinkled arc
7238:
7213:Paley–Wiener
6883:Balanced set
6857:Distribution
6795:Applications
6648:Krein–Milman
6633:Closed graph
6424:
6420:
6393:
6389:
6360:
6356:
6332:
6328:
6325:Schwartz, J.
6309:
6305:
6275:(1): 20–63,
6272:
6268:
6249:
6245:
6222:
6218:
6183:
6179:
6147:
6143:
6114:
6110:
6073:
6054:
6050:
5988:
5984:
5980:
5974:
5933:
5745:
5741:
5737:
5733:
5729:
5725:
5721:
5717:
5678:
5676:
5666:
5663:
5353:
5286:there exist
5188:
5187:is called a
5154:
5149:
4993:
4989:
4985:
4981:
4979:
4596:
4502:
3818:
3236:
3234:
2542:
2540:
2530:
2526:
2524:
2488:
2323:
2319:
2317:
2309:
2085:
2080:
2076:
2071:
2070:
2065:
2051:
2044:
2040:
2032:
1913:
1909:
1904:
1899:
1897:
1892:
1885:
1881:
1874:
1870:
1863:
1859:
1854:
1850:
1846:
1839:
1835:
1828:
1723:
1719:
1715:
1714:is close to
1708:
1699:
1695:
1691:
1685:
1681:
1676:
1672:
1647:
1643:
1638:
1634:
1630:
1625:
1623:
1612:
1608:
1604:
1564:
1563:is an order
1556:
1554:
1543:
1514:
1487:
1483:
1478:
1471:
1426:
1402:
1398:
1394:
1390:
1386:
1382:
1378:
1354:
1350:
1346:
1342:
1118:
695:
220:
172:
155:
152:Introduction
138:Jürgen Moser
126:
120:
105:
96:
89:Please help
85:}}
79:{{
52:
44:
7085:Holomorphic
7068:Directional
7028:Derivatives
6812:Heat kernel
6802:Hardy space
6709:Trace class
6623:Hahn–Banach
6585:Topological
6417:Zehnder, E.
6386:Zehnder, E.
6252:: 499–535,
6225:: 265–315,
6186:(44): 395,
6150:(1): 1–52,
5979:, and that
5748:) given by
5195:if for all
1862:, and then
1595:must be in
175:Nash (1956)
7366:Categories
6745:C*-algebra
6560:Properties
6396:: 91–140,
6265:Nash, John
6107:(PDF-12MB)
6039:References
5973:[0,∞) → Σ(
5326:such that
5076:such that
5022:there are
4362:for fixed
4309:to be the
4254:for fixed
4201:to be the
4119:to be the
3902:maps
3761:for every
3598:for every
3483:such that
3349:such that
3006:such that
2947:and every
2624:such that
2304:See also:
199:KAM theory
7208:Regulated
7180:Integrals
6719:Unbounded
6714:Transpose
6672:Operators
6601:Separable
6596:Reflexive
6581:Algebraic
6567:Barrelled
6240:(1966b),
6213:(1966a),
6172:117923577
6019:∞
5918:∞
5829:−
5824:∞
5803:θ
5781:θ
5698:→
5692:×
5646:×
5643:⋯
5640:×
5634:×
5628:∈
5607:…
5537:‖
5523:‖
5517:⋯
5499:‖
5485:‖
5467:‖
5460:‖
5440:≤
5405:…
5268:→
5262:×
5259:⋯
5256:×
5250:×
5206:∈
5172:→
5109:ε
5103:‖
5090:−
5084:‖
5058:ε
5033:∈
5007:∈
4941:−
4906:θ
4870:Σ
4867:→
4855:Σ
4843:θ
4761:∞
4720:∞
4717:−
4689:→
4626:Σ
4484:→
4424:∞
4342:-norm of
4291:‖
4284:‖
4262:α
4234:-norm of
4220:α
4183:‖
4176:‖
4146:-norm of
4101:‖
4094:‖
4060:∞
3996:∈
3985:∞
3973:‖
3959:‖
3936:∈
3917:→
3879:Σ
3871:that is,
3827:Σ
3790:Σ
3787:∈
3743:‖
3729:‖
3694:∈
3673:≤
3664:‖
3635:‖
3609:∈
3574:‖
3567:‖
3554:≤
3545:‖
3522:‖
3499:∈
3372:→
3360:∘
3334:→
3322:Σ
3287:Σ
3284:→
3206:‖
3199:−
3186:‖
3181:∞
3178:→
3124:∈
3098:ε
3086:‖
3072:−
3059:‖
3037:ε
2992:ε
2955:ε
2935:…
2888:∈
2830:…
2774:‖
2767:‖
2744:∈
2712:∈
2689:⋯
2686:≤
2677:‖
2670:‖
2667:≤
2658:‖
2651:‖
2648:≤
2639:‖
2632:‖
2607:→
2592:‖
2587:⋅
2583:‖
2504:−
2469:→
2463:×
2437:→
2395:∈
2369:→
2337:⊆
2225:−
2125:−
1989:−
1984:∞
1967:θ
1786:−
1781:∞
1650:function
1642:for each
1247:−
1088:∂
1081:β
1073:∂
1054:∂
1047:α
1040:~
1030:∂
1011:α
1007:∑
987:∂
980:β
973:~
963:∂
944:∂
937:α
929:∂
910:α
906:∑
902:↦
896:~
882:given by
850:×
830:Ω
814:→
793:Ω
721:Ω
498:∂
491:α
483:∂
464:∂
457:α
449:∂
430:α
426:∑
392:given by
357:×
341:Ω
321:→
300:Ω
229:Ω
211:Hörmander
7162:Strongly
6963:Analysis
6921:Category
6733:Algebras
6615:Theorems
6572:Complete
6541:Schwartz
6487:glossary
6102:(1982),
6072:(1986).
6051:Mat. Sb.
5760:′
5575:for all
5429:‖
5363:‖
5352:implies
3162:one has
3051:implies
2798:for all
2701:for all
2256:′
2156:′
1858:) is in
207:Hamilton
123:analysis
99:May 2024
7328: (
7270:Related
7222:Results
7198:Dunford
7188:Bochner
7154:Bochner
7128:Measure
6724:Unitary
6704:Nuclear
6689:Compact
6684:Bounded
6679:Adjoint
6653:Min–max
6546:Sobolev
6531:Nuclear
6521:Hilbert
6516:Fréchet
6481: (
6441:0426055
6410:0380867
6379:0418140
6349:0114144
6318:1039945
6297:0075639
6289:1969989
6258:0206461
6231:0199523
6202:0602188
6164:0602181
6133:0656198
6092:0864505
6063:0310924
4834:define
2072:Remark.
1397:, then
197:in the
169:History
7330:bundle
7158:Weakly
7147:Vector
6699:Normal
6536:Orlicz
6526:Hölder
6506:Banach
6495:Spaces
6483:topics
6439:
6408:
6377:
6347:
6316:
6295:
6287:
6256:
6229:
6200:
6170:
6162:
6131:
6090:
6080:
6061:
5732:) and
1891:is in
1884:, and
1880:is in
1869:is in
1571:is in
1481:(Ω;Sym
1433:then |
1377:, and
1305:where
203:Gromov
125:, the
7051:Total
6511:Besov
6285:JSTOR
6168:S2CID
4276:take
4168:take
4086:take
3819:Here
2842:then
2062:says
2038:with
1827:then
1722:) in
1575:then
1513:)) →
1457:with
1393:) is
651:with
187:1966b
183:1966a
6859:(or
6577:Dual
6078:ISBN
5905:and
5720:and
5677:Let
5337:>
5306:and
5106:<
5061:>
5050:and
4984:and
4980:Let
3982:<
3627:and
3441:>
3410:and
3308:and
3095:<
3023:>
2958:>
2322:and
2318:Let
2050:) =
1569:P(f)
1465:and
82:math
6965:in
6429:doi
6398:doi
6365:doi
6337:doi
6277:doi
6188:doi
6152:doi
6119:doi
5736:=Σ(
5728:=Σ(
4891:by
4453:If
4386:If
4027:If
3929:sup
3687:sup
3492:sup
3171:lim
2870:if
2733:if
1613:not
1517:(Ω;
1349:),
700:to
693:."
349:Sym
193:in
144:on
55:vs
7368::
7160:/
7156:/
6485:–
6437:MR
6435:,
6425:29
6423:,
6406:MR
6404:,
6394:28
6392:,
6375:MR
6373:,
6359:,
6345:MR
6343:,
6333:13
6331:,
6314:MR
6310:35
6308:,
6293:MR
6291:,
6283:,
6273:63
6271:,
6254:MR
6250:20
6248:,
6244:,
6227:MR
6223:20
6221:,
6217:,
6198:MR
6196:,
6184:65
6182:,
6166:,
6160:MR
6158:,
6148:62
6146:,
6129:MR
6127:,
6109:,
6088:MR
6086:.
6059:MR
6055:88
5152:.
4594:.
3218:0.
2541:A
2314::
2077:DP
1698:→
1694::
1684:→
1680::
1673:DP
1637:→
1633::
1469:.
1439:−
209:,
205:,
185:,
7332:)
6955:e
6948:t
6941:v
6863:)
6587:)
6583:/
6579:(
6489:)
6471:e
6464:t
6457:v
6431::
6400::
6367::
6361:5
6339::
6300:.
6279::
6190::
6154::
6121::
6115:7
6094:.
6024:.
6015:g
6011:=
6008:)
6005:f
6002:(
5999:P
5989:t
5985:t
5983:(
5981:f
5977:)
5975:B
5959:0
5956:=
5953:)
5950:0
5947:(
5944:f
5934:C
5914:g
5893:0
5890:=
5887:)
5884:0
5881:(
5878:P
5858:.
5853:)
5846:)
5841:)
5838:f
5835:(
5832:P
5820:g
5814:(
5807:t
5799:,
5796:)
5793:f
5790:(
5785:t
5775:(
5770:S
5767:c
5764:=
5757:f
5746:B
5742:c
5738:C
5734:G
5730:B
5726:F
5722:G
5718:F
5704:.
5701:F
5695:G
5689:U
5679:S
5667:r
5661:.
5649:F
5637:F
5631:U
5624:)
5618:k
5614:h
5610:,
5604:,
5599:1
5595:h
5591:,
5588:f
5584:(
5560:)
5555:1
5552:+
5547:r
5544:+
5541:n
5531:k
5527:h
5520:+
5514:+
5509:r
5506:+
5503:n
5493:1
5489:h
5482:+
5477:r
5474:+
5471:n
5463:f
5455:(
5448:n
5444:C
5435:n
5422:)
5416:k
5412:h
5408:,
5402:,
5397:1
5393:h
5389:,
5386:f
5382:(
5378:P
5373:k
5369:D
5340:b
5334:n
5314:b
5294:r
5271:G
5265:F
5253:F
5247:U
5244::
5241:P
5236:k
5232:D
5210:N
5203:k
5175:G
5169:U
5166::
5163:P
5150:U
5134:1
5130:f
5098:1
5094:f
5087:f
5064:0
5037:N
5030:n
5010:U
5004:f
4994:F
4990:U
4986:G
4982:F
4960:.
4955:i
4951:x
4947:)
4944:i
4938:t
4935:(
4932:s
4929:=
4924:i
4919:)
4915:x
4910:t
4901:(
4879:)
4876:B
4873:(
4864:)
4861:B
4858:(
4852::
4847:t
4822:t
4802:.
4799:]
4796:1
4793:,
4790:0
4787:[
4767:,
4764:)
4758:,
4755:1
4752:(
4732:,
4729:)
4726:0
4723:,
4714:(
4693:R
4685:R
4681::
4678:s
4658:,
4655:B
4635:)
4632:B
4629:(
4606:B
4578:L
4556:1
4552:L
4531:B
4511:M
4487:M
4481:V
4461:M
4438:,
4435:)
4432:M
4429:(
4419:0
4415:C
4394:M
4370:p
4350:f
4328:p
4325:,
4322:n
4318:W
4295:n
4287:f
4242:f
4217:,
4214:n
4210:C
4187:n
4179:f
4154:f
4132:n
4128:C
4105:n
4097:f
4071:)
4068:M
4065:(
4056:C
4035:M
4011:.
4006:}
4000:N
3993:n
3977:B
3967:k
3963:x
3954:k
3951:n
3947:e
3940:N
3933:k
3920:B
3913:N
3909::
3906:x
3896:{
3891:=
3888:)
3885:B
3882:(
3859:,
3856:B
3836:)
3833:B
3830:(
3802:.
3799:)
3796:B
3793:(
3783:}
3778:i
3774:x
3770:{
3747:B
3737:k
3733:x
3724:k
3721:)
3718:n
3715:+
3712:r
3709:(
3705:e
3698:N
3691:k
3681:n
3677:C
3668:n
3660:)
3657:}
3652:i
3648:x
3644:{
3641:(
3638:M
3615:,
3612:F
3606:f
3584:n
3581:+
3578:r
3570:f
3562:n
3558:C
3549:B
3539:k
3535:)
3531:f
3528:(
3525:L
3517:k
3514:n
3510:e
3503:N
3496:k
3469:n
3465:C
3444:b
3438:n
3418:b
3398:r
3375:F
3369:F
3366::
3363:L
3357:M
3337:F
3331:)
3328:B
3325:(
3319::
3316:M
3296:)
3293:B
3290:(
3281:F
3278::
3275:L
3255:B
3215:=
3210:n
3202:f
3194:j
3190:f
3175:j
3150:,
3147:n
3127:F
3121:f
3101:,
3090:n
3080:k
3076:f
3067:j
3063:f
3034:,
3031:n
3027:N
3020:k
3017:,
3014:j
2989:,
2986:n
2982:N
2961:0
2932:,
2929:2
2926:,
2923:1
2920:,
2917:0
2914:=
2911:n
2891:F
2883:j
2879:f
2856:0
2853:=
2850:f
2827:,
2824:2
2821:,
2818:1
2815:,
2812:0
2809:=
2806:n
2786:0
2783:=
2778:n
2770:f
2747:F
2741:f
2718:.
2715:F
2709:f
2681:2
2673:f
2662:1
2654:f
2643:0
2635:f
2611:R
2604:F
2601::
2596:n
2561:F
2531:P
2527:P
2507:1
2500:P
2489:P
2475:,
2472:F
2466:G
2460:U
2440:G
2434:F
2431::
2426:f
2422:P
2418:d
2398:U
2392:f
2372:G
2366:U
2363::
2360:P
2340:F
2334:U
2324:G
2320:F
2279:,
2273:)
2268:0
2264:x
2260:(
2253:f
2247:)
2242:n
2238:x
2234:(
2231:f
2220:n
2216:x
2212:=
2207:1
2204:+
2201:n
2197:x
2173:)
2168:n
2164:x
2160:(
2153:f
2147:)
2142:n
2138:x
2134:(
2131:f
2120:n
2116:x
2112:=
2107:1
2104:+
2101:n
2097:x
2086:f
2081:f
2055:∞
2052:g
2048:∞
2045:f
2043:(
2041:P
2036:∞
2033:f
2016:)
2011:)
2008:)
2003:n
1999:f
1995:(
1992:P
1980:g
1976:(
1971:n
1961:(
1956:S
1953:+
1948:n
1944:f
1940:=
1935:1
1932:+
1929:n
1925:f
1914:n
1910:C
1905:n
1900:n
1893:C
1889:4
1886:f
1882:C
1878:3
1875:f
1871:C
1867:2
1864:f
1860:C
1855:n
1851:f
1849:(
1847:P
1845:−
1843:∞
1840:g
1836:C
1834:∈
1832:1
1829:f
1815:,
1810:)
1805:)
1800:n
1796:f
1792:(
1789:P
1777:g
1771:(
1766:S
1763:+
1758:n
1754:f
1750:=
1745:1
1742:+
1739:n
1735:f
1724:C
1720:f
1718:(
1716:P
1712:∞
1709:g
1700:C
1696:C
1692:S
1686:C
1682:C
1677:f
1658:f
1648:C
1644:k
1639:C
1635:C
1631:P
1626:P
1609:C
1605:C
1597:C
1583:f
1573:C
1565:k
1561:P
1550:P
1546:C
1539:)
1526:R
1515:C
1500:R
1491:(
1488:n
1486:×
1484:n
1479:C
1474:P
1467:Q
1463:P
1459:P
1455:Q
1451:Q
1447:C
1443:h
1441:|
1437:|
1435:H
1431:C
1427:C
1413:f
1403:C
1399:R
1395:C
1391:f
1389:(
1387:P
1383:f
1381:(
1379:h
1365:f
1355:f
1353:(
1351:H
1347:f
1345:(
1343:P
1327:)
1324:f
1321:(
1318:P
1314:R
1293:,
1288:2
1283:)
1280:f
1277:(
1274:P
1269:|
1264:)
1261:f
1258:(
1255:h
1251:|
1242:2
1237:|
1232:)
1229:f
1226:(
1223:H
1219:|
1215:=
1210:)
1207:f
1204:(
1201:P
1197:R
1176:f
1156:f
1136:)
1133:f
1130:(
1127:P
1104:.
1096:j
1092:u
1077:f
1062:i
1058:u
1037:f
1022:N
1017:1
1014:=
1003:+
995:j
991:u
970:f
952:i
948:u
933:f
921:N
916:1
913:=
893:f
869:)
866:)
862:R
858:(
853:n
847:n
843:m
839:y
836:S
833:;
827:(
822:4
818:C
811:)
806:N
801:R
796:;
790:(
785:5
781:C
759:f
739:)
734:N
729:R
724:;
718:(
713:5
709:C
698:P
681:g
678:=
675:)
670:g
666:f
662:(
659:P
637:g
633:f
612:)
609:f
606:(
603:P
583:g
563:)
560:f
557:(
554:P
534:f
514:.
506:j
502:u
487:f
472:i
468:u
453:f
441:N
436:1
433:=
422:=
417:j
414:i
410:)
406:f
403:(
400:P
378:)
373:)
369:R
365:(
360:n
354:n
344:;
336:(
329:0
325:C
318:)
313:N
308:R
303:;
297:(
292:1
288:C
284::
281:P
271:.
257:n
252:R
112:)
106:(
101:)
97:(
93:.
64:R
53:R
49:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.