4480:
957:
689:
783:, shown on the figure at the right, satisfies all Kakutani's conditions, and indeed it has many fixed points: any point on the 45° line (dotted line in red) which intersects the graph of the function (shaded in grey) is a fixed point, so in fact there is an infinity of fixed points in this particular case. For example,
3221:-simplex is the higher-dimensional version of a triangle. Proving Kakutani's theorem for set-valued function defined on a simplex is not essentially different from proving it for intervals. The additional complexity in the higher-dimensional case exists in the first step of chopping up the domain into finer subpieces:
2315:
Thus, the closed intervals form a sequence of subintervals of . Condition (2) tells us that these subintervals continue to become smaller while condition (3)–(6) tell us that the function φ shifts the left end of each subinterval to its right and shifts the right end of each subinterval to its left.
1619:
does not equate supply and demand everywhere. The challenge here is to construct φ so that it has this property while at the same time satisfying the conditions in
Kakutani's theorem. If this can be done then φ has a fixed point according to the theorem. Given the way it was constructed, this fixed
2996:
Otherwise, we can write the following. Recall that we can parameterize a line between two points a and b by (1-t)a + tb. Using our finding above that q<x<p, we can create such a line between p and q as a function of x (notice the fractions below are on the unit interval). By a convenient
1133:
3473:
recalls that
Kakutani once asked him at a conference why so many economists had attended his talk. When Binmore told him that it was probably because of the Kakutani fixed point theorem, Kakutani was puzzled and replied, "What is the Kakutani fixed point theorem?"
1584:
theory in economics, Kakutani's theorem has been used to prove the existence of set of prices which simultaneously equate supply with demand in all markets of an economy. The existence of such prices had been an open question in economics going back to at least
3170:
935:
3259:. Once again we employ the same technique of creating increasingly finer subdivisions. But instead of triangles with straight edges as in the case of n-simplices, we now use triangles with curved edges. In formal terms, we find a simplex which covers
1258:
1557:) is nonempty since there is always at least one best response. It is convex, since a mixture of two best-responses for a player is still a best-response for the player. It can be proved that φ has a closed graph.
1564:
of the game is defined as a fixed point of φ, i.e. a tuple of strategies where each player's strategy is a best response to the strategies of the other players. Kakutani's theorem ensures that this fixed point
3980:
theory. Chapter 5 uses
Kakutani's theorem to prove the existence of equilibrium prices. Appendix C includes a proof of Kakutani's theorem and discusses its relationship with other mathematical results used in
1781:
2055:
3244:
Once these changes have been made to the first step, the second and third steps of finding a limiting point and proving that it is a fixed point are almost unchanged from the one-dimensional case.
3232:
While in the one-dimensional case we could use elementary arguments to pick one of the half-intervals in a way that its end-points were moved in opposite directions, in the case of simplices the
985:
469:
421:
3011:
801:
1949:
584:
325:, which is used to refer to a function that for each input may return many outputs. Thus, each element of the domain corresponds to a subset of one or more elements of the range.
1864:
639:
4369:
1703:
1156:
781:
2005:
535:
502:
319:
1808:
1655:
Brouwer's fixed-point theorem is a special case of
Kakutani fixed-point theorem. Conversely, Kakutani fixed-point theorem is an immediate generalization via the
4032:
1972:
1263:
The function has no fixed point. Though it satisfies all other requirements of
Kakutani's theorem, its graph is not closed; for example, consider the sequences
604:
1437:
is required to be closed-valued in the alternative statement of the
Kakutani theorem, the Closed Graph Theorem implies that the two statements are equivalent.
4195:
2089:
of the real line. Moreover, the proof of this case is instructive since its general strategy can be carried over to the higher-dimensional case as well.
4322:
4177:
4153:
3445:
3405:
3280:
3968:
3903:
3874:
3726:
3664:
3947:(Comprehensive high-level mathematical treatment of fixed point theory, including the infinite dimensional analogues of Kakutani's theorem.)
4509:
4519:
4045:
4134:
4025:
3597:
1138:
The function has no fixed point. Though it satisfies all other requirements of
Kakutani's theorem, its value fails to be convex at
4404:
1708:
4049:
2010:
3751:
79:
1545:) associates with each tuple a new tuple where each player's strategy is her best response to other players' strategies in
1128:{\displaystyle \varphi (x)={\begin{cases}3/4&0\leq x<0.5\\\{3/4,1/4\}&x=0.5\\1/4&0.5<x\leq 1\end{cases}}}
4514:
4504:
4200:
4003:
3455:
on S which has a closed graph and the property that φ(x) is non-empty and convex for all x ∈ S. Then the set of
3414:
4256:
4483:
4190:
4018:
4220:
1549:. Since there may be a number of responses which are equally good, φ is set-valued rather than single-valued. For each
1390:
This statement of
Kakutani's theorem is completely equivalent to the statement given at the beginning of this article.
3998:
2780:
2685:
71:
4465:
4225:
3165:{\displaystyle x=\left({\frac {x-q^{*}}{p^{*}-q^{*}}}\right)p^{*}+\left(1-{\frac {x-q^{*}}{p^{*}-q^{*}}}\right)q^{*}}
4419:
4343:
426:
378:
4460:
2684:
We have a pair of sequences of intervals, and we would like to show them to converge to a limiting point with the
930:{\displaystyle \varphi (x)={\begin{cases}3/4&0\leq x<0.5\\{}&x=0.5\\1/4&0.5<x\leq 1\end{cases}}}
82:
defined on compact, convex subsets of
Euclidean spaces. Kakutani's theorem extends this to set-valued functions.
4276:
3456:
1382:
219:
63:
4210:
4312:
4113:
3350:
1486:
4185:
1869:
4409:
3226:
540:
4440:
4384:
4348:
3680:
McKenzie, Lionel (1954). "On
Equilibrium in Graham's Model of World Trade and Other Competitive Systems".
2733:
2086:
1636:
1485:
in every finite game with mixed strategies for any finite number of players. This work later earned him a
1446:
264:
1813:
90:
31:
3284:
1530:
is the cartesian product of all these simplices. It is indeed a nonempty, compact and convex subset of
3993:
4423:
3806:
3542:
3304:
3255:
Kakutani's theorem for n-simplices can be used to prove the theorem for an arbitrary compact, convex
1394:
1253:{\displaystyle \varphi (x)={\begin{cases}3/4&0\leq x<0.5\\1/4&0.5\leq x\leq 1\end{cases}}}
47:
3752:"A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium"
1180:
1009:
825:
4389:
4327:
4041:
3977:
3452:
3268:
2773:
2093:
1644:
1581:
1575:
1344:
609:
163:
43:
39:
3932:(Standard reference on fixed-point theory for economists. Includes a proof of Kakutani's theorem.)
1669:
1635:
Kakutani's fixed-point theorem is used in proving the existence of cake allocations that are both
708:
4414:
4281:
3771:
3699:
3237:
1977:
507:
474:
4394:
3964:
3899:
3891:
3870:
3834:
3732:
3722:
3660:
3593:
3570:
2725:
1656:
1630:
67:
4399:
4317:
4286:
4266:
4251:
4246:
4241:
4078:
3824:
3814:
3763:
3691:
3614:
3560:
3550:
3526:
3505:
1640:
1561:
1514:
1482:
1474:
372:
298:
94:
17:
3420:
There is another version that the statement of the theorem becomes the same as that in the
1786:
4261:
4215:
4163:
4158:
4129:
4010:
3493:
3448:
3421:
3402:
2347:
be any point in φ(1). Then, conditions (1)–(4) are immediately fulfilled. Moreover, since
2096:
on the closed interval which satisfies the conditions of Kakutani's fixed-point theorem.
1590:
1463:
1426:
1398:
1320:
140:
86:
59:
4088:
2101:
Create a sequence of subdivisions of with adjacent points moving in opposite directions.
3810:
3546:
4450:
4302:
4103:
3957:
3937:
3857:
3829:
3794:
3565:
3530:
1957:
1586:
1502:
1341:
1290:
589:
2688:. To do so, we construe these two interval sequences as a single sequence of points, (
4498:
4455:
4379:
4108:
4093:
4083:
3952:
3233:
1615:) is chosen so that its result differs from its arguments as long as the price-tuple
1467:
70:
to a set containing it. The Kakutani fixed point theorem is a generalization of the
4445:
4098:
4068:
3682:
1620:
point must correspond to a price-tuple which equates supply with demand everywhere.
1430:
180:
3779:
3509:
3217:
are the simplest objects on which Kakutani's theorem can be proved. Informally, a
2361:≥ 0 and hence condition (5) is fulfilled. Similarly condition (6) is fulfilled by
2085:
The proof of Kakutani's theorem is simplest for set-valued functions defined over
3861:
3654:
787: = 0.72 (dashed line in blue) is a fixed point since 0.72 ∈ .
4374:
4364:
4271:
4073:
3470:
3434:
3391:
2776:
2729:
1513:-tuple of probabilities summing up to 1, so each player's strategy space is the
1478:
1457:
1309:
129:
98:
55:
4307:
4147:
4143:
4139:
3922:
3795:"Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces"
3650:
3438:
3395:
3002:
2441:
1313:
360:
133:
51:
3736:
3225:
Where we split intervals into two at the middle in the one-dimensional case,
3430:
3387:
1666:
By the approximate selection theorem, there exists a sequence of continuous
1470:. This application was specifically discussed by Kakutani's original paper.
1305:
272:
125:
102:
3838:
3574:
3819:
956:
3555:
3309:
2147:
75:
3775:
3703:
3211:
1289:
Some sources, including Kakutani's original paper, use the concept of
940:
satisfies all Kakutani's conditions, and indeed it has a fixed point:
3441:
3398:
3291:. To state the theorem in this case, we need a few more definitions:
3288:
1316:
136:
3767:
3695:
3279:
Kakutani's fixed-point theorem was extended to infinite-dimensional
3271:. Then we can apply the already established result for n-simplices.
1413:→2 has a closed graph if and only if it is upper hemicontinuous and
688:
3927:
Fixed Point Theorems with Applications to Economics and Game Theory
3590:
Fixed Point Theorems with Applications to Economics and Game Theory
2783:. Let's fix attention on such a subsequence and let its limit be (
1605:
1498:
1477:
used the Kakutani fixed point theorem to prove a major result in
3240:
is used to guarantee the existence of an appropriate subsimplex.
4014:
3408:. Let φ: S→2 be a Kakutani map. Then φ has a fixed point.
3496:(1941). "A generalization of Brouwer's fixed point theorem".
74:. The Brouwer fixed point theorem is a fundamental result in
3635:
Aliprantis, Charlambos; Kim C. Border (1999). "Chapter 17".
3413:
The corresponding result for single-valued functions is the
2799:*). Since the graph of φ is closed it must be the case that
1481:. Stated informally, the theorem implies the existence of a
3381:
Then the Kakutani–Glicksberg–Fan theorem can be stated as:
1776:{\displaystyle \operatorname {graph} (f_{n})\subset _{1/n}}
1246:
1121:
923:
1783:. By Brouwer fixed-point theorem, there exists a sequence
1462:
The Kakutani fixed point theorem can be used to prove the
4205:
3229:
is used to break up a simplex into smaller sub-simplices.
2050:{\displaystyle (x,x)\in \operatorname {graph} (\varphi )}
1397:
for set-valued functions, which says that for a compact
3898:(1st ed.). Oxford University Press. p. 256.
97:. It has subsequently found widespread application in
3721:. Springer International Publishing. pp. 68–70.
3014:
2013:
1980:
1960:
1872:
1816:
1789:
1711:
1672:
1159:
988:
804:
711:
612:
592:
543:
510:
477:
429:
381:
301:
3860:; Andrzej Granas (2003). "Chapter II, Section 5.8".
1589:. The first proof of this result was constructed by
1505:
chosen by each player in a game. If each player has
4433:
4357:
4336:
4295:
4234:
4176:
4122:
4057:
3637:
Infinite Dimensional Analysis: A Hitchhiker's Guide
2319:Such a sequence can be constructed as follows. Let
1509:possible actions, then each player's strategy is a
4370:Spectral theory of ordinary differential equations
3956:
3164:
2049:
1999:
1966:
1943:
1858:
1802:
1775:
1697:
1252:
1127:
929:
775:
633:
598:
578:
529:
496:
463:
415:
313:
1974:is compact, we can take a convergent subsequence
3756:Proceedings of the American Mathematical Society
78:which proves the existence of fixed points for
2956:Show that the limiting point is a fixed point.
651: → 2 be a set-valued function. Then
263:. Formally it can be seen just as an ordinary
4026:
1150:Consider the following function defined on :
1146:A function that does not satisfy closed graph
979:Consider the following function defined on :
464:{\displaystyle \{y_{n}\}_{n\in \mathbb {N} }}
416:{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}
8:
3630:
3628:
1072:
1044:
701:A function with infinitely many fixed points
444:
430:
396:
382:
27:Fixed-point theorem for set-valued functions
3852:
3850:
3848:
3369:) is non-empty, compact and convex for all
4061:
4033:
4019:
4011:
3488:
3486:
2679:Find a limiting point of the subdivisions.
952:A function that does not satisfy convexity
3828:
3818:
3564:
3554:
3156:
3138:
3125:
3113:
3100:
3080:
3063:
3050:
3038:
3025:
3013:
2012:
1985:
1979:
1959:
1931:
1927:
1893:
1880:
1871:
1850:
1834:
1821:
1815:
1794:
1788:
1763:
1759:
1725:
1710:
1677:
1671:
1651:Relation to Brouwer's fixed-point theorem
1218:
1186:
1175:
1158:
1093:
1064:
1050:
1015:
1004:
987:
895:
860:
831:
820:
803:
762:
739:
710:
611:
591:
567:
548:
542:
515:
509:
482:
476:
455:
454:
447:
437:
428:
407:
406:
399:
389:
380:
300:
4323:Group algebra of a locally compact group
3281:locally convex topological vector spaces
2772:) lies in a compact set, it must have a
2154:
1364:is non-empty, closed, and convex for all
955:
687:
3896:Playing for Real: A Text on Game Theory
3482:
3406:locally convex topological vector space
1944:{\displaystyle (x_{n},x_{n})\in _{1/n}}
50:for a set-valued function defined on a
3531:"Equilibrium Points in N-Person Games"
3521:
3519:
976:is essential for the theorem to hold.
579:{\displaystyle y_{n}\in \phi (x_{n})}
7:
3365:if it is upper hemicontinuous and φ(
3275:Infinite-dimensional generalizations
2354:∈ φ(0) ⊂ , it must be the case that
791:A function with a unique fixed point
1650:
1859:{\displaystyle f_{n}(x_{n})=x_{n}}
25:
335: → 2 is said to have a
251:is some rule that associates one
4479:
4478:
4405:Topological quantum field theory
3892:"When Do Nash Equilibria Exist?"
3357:→2 be a set-valued function. If
2926:Then we have the situation that
2815:*). Moreover, by condition (5),
2542:) is non-empty, there must be a
3263:and then move the problem from
2150:with the following properties:
960:A function without fixed points
948:is contained in the interval .
197:is non-empty and convex for all
3659:. Cambridge University Press.
3592:. Cambridge University Press.
3459:of φ is non-empty and compact.
3361:is convex, then φ is termed a
3298:A set-valued function φ:
2044:
2038:
2026:
2014:
1991:
1924:
1920:
1914:
1905:
1899:
1873:
1840:
1827:
1756:
1752:
1746:
1737:
1731:
1718:
1689:
1393:We can show this by using the
1169:
1163:
998:
992:
944:= 0.5 is a fixed point, since
874:
862:
814:
808:
770:
727:
721:
715:
628:
622:
573:
560:
521:
488:
331:A set-valued function φ:
172:with the following properties:
1:
4201:Uniform boundedness principle
3929:. Cambridge University Press.
3510:10.1215/S0012-7094-41-00838-4
3469:In his game theory textbook,
3415:Tychonoff fixed-point theorem
1657:approximate selection theorem
634:{\displaystyle y\in \phi (x)}
85:The theorem was developed by
3959:General Competitive Analysis
3535:Proc. Natl. Acad. Sci. U.S.A
2674:satisfy conditions (1)–(6).
1698:{\displaystyle f_{n}:S\to S}
776:{\displaystyle \varphi (x)=}
36:Kakutani fixed-point theorem
18:Kakutani fixed point theorem
4510:Theorems in convex geometry
3999:Encyclopedia of Mathematics
3210:In dimensions greater one,
3175:it once again follows that
2686:Bolzano-Weierstrass theorem
2371:Now suppose we have chosen
1293:while stating the theorem:
113:Kakutani's theorem states:
72:Brouwer fixed point theorem
4536:
4520:General equilibrium theory
4344:Invariant subspace problem
3656:General Equilibrium Theory
3267:to the simplex by using a
2997:writing of x, and since φ(
2057:since it is a closed set.
2000:{\displaystyle x_{n}\to x}
1643:. This result is known as
1628:
1573:
1455:
1444:
1421:) is a closed set for all
530:{\displaystyle y_{n}\to y}
497:{\displaystyle x_{n}\to x}
287: → 2, such that
4474:
4064:
3940:; Andrzej Granas (2003).
3750:Glicksberg, I.L. (1952).
3717:Shapiro, Joel H. (2016).
3639:(3rd ed.). Springer.
3498:Duke Mathematical Journal
3351:topological vector spaces
2407:satisfying (1)–(6). Let,
2340:be any point in φ(0) and
2186:
2162:
295:) is non-empty for every
89:in 1941, and was used by
4313:Spectrum of a C*-algebra
3799:Proc Natl Acad Sci U S A
2852:) ≤ 2 by condition (2),
2823:* and by condition (6),
2634:It can be verified that
1487:Nobel Prize in Economics
1405:, a set-valued function
321:. Some prefer the term
66:, i.e. a point which is
4410:Noncommutative geometry
3976:(Standard reference on
3619:A Course in Game Theory
3588:Border, Kim C. (1989).
3451:. Let φ: S→2 be a
3227:barycentric subdivision
3195:is a fixed point of φ.
2993:is a fixed point of φ.
375:i.e. for all sequences
4466:Tomita–Takesaki theory
4441:Approximation property
4385:Calculus of variations
3782:on September 22, 2017.
3166:
2736:. Since our sequence (
2051:
2001:
1968:
1945:
1860:
1804:
1777:
1699:
1447:Mathematical economics
1353:with the property that
1254:
1129:
961:
931:
777:
697:
635:
600:
580:
531:
498:
465:
417:
315:
314:{\displaystyle x\in X}
93:in his description of
4461:Banach–Mazur distance
4424:Generalized functions
3955:; F. H. Hahn (1971).
3890:Binmore, Ken (2007).
3820:10.1073/pnas.38.2.121
3719:A Fixed-Point Farrago
3621:. Cambridge, MA: MIT.
3167:
2052:
2002:
1969:
1946:
1861:
1805:
1803:{\displaystyle x_{n}}
1778:
1700:
1429:are Hausdorff (being
1285:Alternative statement
1255:
1130:
964:The requirement that
959:
932:
778:
691:
636:
601:
581:
532:
499:
466:
418:
316:
48:sufficient conditions
32:mathematical analysis
4515:Theorems in topology
4505:Fixed-point theorems
4206:Kakutani fixed-point
4191:Riesz representation
3613:Osborne, Martin J.;
3556:10.1073/pnas.36.1.48
3305:upper hemicontinuous
3295:Upper hemicontinuity
3012:
2724:). This lies in the
2558:. In this case let,
2011:
1978:
1958:
1870:
1814:
1787:
1709:
1670:
1608:of commodity prices.
1395:closed graph theorem
1342:upper hemicontinuous
1291:upper hemicontinuity
1157:
986:
972:) be convex for all
802:
709:
610:
590:
541:
508:
475:
427:
379:
299:
80:continuous functions
44:set-valued functions
4390:Functional calculus
4349:Mahler's conjecture
4328:Von Neumann algebra
4042:Functional analysis
3978:general equilibrium
3811:1952PNAS...38..121F
3547:1950PNAS...36...48N
3453:set-valued function
3269:deformation retract
2781:Bolzano-Weierstrass
2734:Tychonoff's theorem
2538:Otherwise, since φ(
2094:set-valued function
1582:general equilibrium
1576:General equilibrium
1570:General equilibrium
1345:set-valued function
259:with each point in
238:set-valued function
233:Set-valued function
164:set-valued function
40:fixed-point theorem
4415:Riemann hypothesis
4114:Topological vector
3994:"Kakutani theorem"
3942:Fixed Point Theory
3863:Fixed Point Theory
3162:
2047:
1997:
1964:
1941:
1856:
1800:
1773:
1695:
1664:
1250:
1245:
1125:
1120:
962:
927:
922:
773:
698:
631:
596:
576:
527:
494:
461:
413:
311:
4492:
4491:
4395:Integral operator
4172:
4171:
3982:
3970:978-0-8162-0275-1
3953:Arrow, Kenneth J.
3948:
3933:
3905:978-0-19-804114-6
3876:978-0-387-00173-9
3867:(limited preview)
3728:978-3-319-27978-7
3666:978-0-521-56473-1
3615:Rubinstein, Ariel
3285:Irving Glicksberg
3179:must belong to φ(
3145:
3070:
2726:cartesian product
2311:
2310:
1967:{\displaystyle S}
1662:
1631:Fair cake-cutting
1466:in the theory of
752:
692:Fixed points for
599:{\displaystyle n}
16:(Redirected from
4527:
4482:
4481:
4400:Jones polynomial
4318:Operator algebra
4062:
4035:
4028:
4021:
4012:
4007:
3975:
3974:
3962:
3946:
3945:
3931:
3930:
3910:
3909:
3887:
3881:
3880:
3868:
3854:
3843:
3842:
3832:
3822:
3793:Fan, Ky (1952).
3790:
3784:
3783:
3778:. Archived from
3747:
3741:
3740:
3714:
3708:
3707:
3677:
3671:
3670:
3647:
3641:
3640:
3632:
3623:
3622:
3610:
3604:
3603:
3585:
3579:
3578:
3568:
3558:
3523:
3514:
3513:
3494:Kakutani, Shizuo
3490:
3236:result known as
3171:
3169:
3168:
3163:
3161:
3160:
3151:
3147:
3146:
3144:
3143:
3142:
3130:
3129:
3119:
3118:
3117:
3101:
3085:
3084:
3075:
3071:
3069:
3068:
3067:
3055:
3054:
3044:
3043:
3042:
3026:
2728:×××, which is a
2463:, then we take,
2155:
2087:closed intervals
2082:
2081:
2077:
2056:
2054:
2053:
2048:
2006:
2004:
2003:
1998:
1990:
1989:
1973:
1971:
1970:
1965:
1950:
1948:
1947:
1942:
1940:
1939:
1935:
1898:
1897:
1885:
1884:
1865:
1863:
1862:
1857:
1855:
1854:
1839:
1838:
1826:
1825:
1809:
1807:
1806:
1801:
1799:
1798:
1782:
1780:
1779:
1774:
1772:
1771:
1767:
1730:
1729:
1704:
1702:
1701:
1696:
1682:
1681:
1645:Weller's theorem
1641:Pareto efficient
1562:Nash equilibrium
1515:standard simplex
1503:mixed strategies
1489:. In this case:
1483:Nash equilibrium
1427:Euclidean spaces
1259:
1257:
1256:
1251:
1249:
1248:
1222:
1190:
1134:
1132:
1131:
1126:
1124:
1123:
1097:
1068:
1054:
1019:
936:
934:
933:
928:
926:
925:
899:
861:
835:
782:
780:
779:
774:
766:
750:
743:
640:
638:
637:
632:
605:
603:
602:
597:
585:
583:
582:
577:
572:
571:
553:
552:
536:
534:
533:
528:
520:
519:
503:
501:
500:
495:
487:
486:
470:
468:
467:
462:
460:
459:
458:
442:
441:
422:
420:
419:
414:
412:
411:
410:
394:
393:
373:product topology
320:
318:
317:
312:
21:
4535:
4534:
4530:
4529:
4528:
4526:
4525:
4524:
4495:
4494:
4493:
4488:
4470:
4434:Advanced topics
4429:
4353:
4332:
4291:
4257:Hilbert–Schmidt
4230:
4221:Gelfand–Naimark
4168:
4118:
4053:
4039:
3992:
3989:
3971:
3951:
3938:Dugundji, James
3936:
3921:
3918:
3916:Further reading
3913:
3906:
3889:
3888:
3884:
3877:
3866:
3858:Dugundji, James
3856:
3855:
3846:
3792:
3791:
3787:
3768:10.2307/2032478
3749:
3748:
3744:
3729:
3716:
3715:
3711:
3696:10.2307/1907539
3679:
3678:
3674:
3667:
3649:
3648:
3644:
3634:
3633:
3626:
3612:
3611:
3607:
3600:
3587:
3586:
3582:
3525:
3524:
3517:
3492:
3491:
3484:
3480:
3467:
3449:Hausdorff space
3277:
3253:
3238:Sperner's lemma
3208:
3191:* do and hence
3152:
3134:
3121:
3120:
3109:
3102:
3093:
3089:
3076:
3059:
3046:
3045:
3034:
3027:
3021:
3010:
3009:
2898:
2889:
2880:
2871:
2851:
2842:
2771:
2762:
2753:
2744:
2723:
2714:
2705:
2696:
2673:
2663:
2653:
2643:
2625:
2614:
2605:
2590:
2579:
2570:
2534:
2525:
2510:
2499:
2490:
2475:
2440:∈ because is
2431:
2422:
2406:
2397:
2388:
2379:
2367:
2360:
2353:
2346:
2339:
2332:
2325:
2307:
2298:
2283:
2274:
2256:
2247:
2231:
2222:
2204:
2195:
2178:
2169:
2146:= 0, 1, … be a
2141:
2132:
2123:
2114:
2092:Let φ: →2 be a
2083:
2079:
2075:
2073:
2072:
2064:
2059:
2009:
2008:
1981:
1976:
1975:
1956:
1955:
1923:
1889:
1876:
1868:
1867:
1846:
1830:
1817:
1812:
1811:
1790:
1785:
1784:
1755:
1721:
1707:
1706:
1673:
1668:
1667:
1653:
1633:
1627:
1611:The function φ(
1591:Lionel McKenzie
1578:
1572:
1541:The function φ(
1464:minimax theorem
1460:
1454:
1449:
1443:
1321:Euclidean space
1287:
1279:
1268:
1244:
1243:
1226:
1212:
1211:
1194:
1176:
1155:
1154:
1148:
1119:
1118:
1101:
1087:
1086:
1075:
1041:
1040:
1023:
1005:
984:
983:
954:
921:
920:
903:
889:
888:
877:
857:
856:
839:
821:
800:
799:
793:
707:
706:
703:
686:
608:
607:
588:
587:
563:
544:
539:
538:
511:
506:
505:
478:
473:
472:
443:
433:
425:
424:
395:
385:
377:
376:
297:
296:
230:
160: → 2
141:Euclidean space
111:
95:Nash equilibria
87:Shizuo Kakutani
60:Euclidean space
28:
23:
22:
15:
12:
11:
5:
4533:
4531:
4523:
4522:
4517:
4512:
4507:
4497:
4496:
4490:
4489:
4487:
4486:
4475:
4472:
4471:
4469:
4468:
4463:
4458:
4453:
4451:Choquet theory
4448:
4443:
4437:
4435:
4431:
4430:
4428:
4427:
4417:
4412:
4407:
4402:
4397:
4392:
4387:
4382:
4377:
4372:
4367:
4361:
4359:
4355:
4354:
4352:
4351:
4346:
4340:
4338:
4334:
4333:
4331:
4330:
4325:
4320:
4315:
4310:
4305:
4303:Banach algebra
4299:
4297:
4293:
4292:
4290:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4238:
4236:
4232:
4231:
4229:
4228:
4226:Banach–Alaoglu
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4188:
4182:
4180:
4174:
4173:
4170:
4169:
4167:
4166:
4161:
4156:
4154:Locally convex
4151:
4137:
4132:
4126:
4124:
4120:
4119:
4117:
4116:
4111:
4106:
4101:
4096:
4091:
4086:
4081:
4076:
4071:
4065:
4059:
4055:
4054:
4040:
4038:
4037:
4030:
4023:
4015:
4009:
4008:
3988:
3987:External links
3985:
3984:
3983:
3969:
3963:. Holden-Day.
3949:
3934:
3923:Border, Kim C.
3917:
3914:
3912:
3911:
3904:
3882:
3875:
3844:
3805:(2): 121–126.
3785:
3762:(1): 170–174.
3742:
3727:
3709:
3690:(2): 147–161.
3672:
3665:
3651:Starr, Ross M.
3642:
3624:
3605:
3598:
3580:
3527:Nash, J.F. Jr.
3515:
3504:(3): 457–459.
3481:
3479:
3476:
3466:
3463:
3462:
3461:
3446:locally convex
3411:
3410:
3379:
3378:
3339:
3336:
3327:) ⊂
3296:
3276:
3273:
3252:
3246:
3242:
3241:
3230:
3207:
3197:
3173:
3172:
3159:
3155:
3150:
3141:
3137:
3133:
3128:
3124:
3116:
3112:
3108:
3105:
3099:
3096:
3092:
3088:
3083:
3079:
3074:
3066:
3062:
3058:
3053:
3049:
3041:
3037:
3033:
3030:
3024:
3020:
3017:
2959:
2958:
2952:
2951:
2901:
2900:
2894:
2885:
2876:
2867:
2847:
2838:
2767:
2758:
2749:
2740:
2719:
2710:
2701:
2692:
2682:
2681:
2668:
2658:
2648:
2638:
2632:
2631:
2620:
2615:
2610:
2600:
2595:
2585:
2580:
2575:
2565:
2536:
2535:
2530:
2520:
2515:
2505:
2500:
2495:
2485:
2480:
2470:
2447:If there is a
2434:
2433:
2427:
2418:
2402:
2393:
2384:
2375:
2365:
2358:
2351:
2344:
2337:
2330:
2323:
2313:
2312:
2309:
2308:
2303:
2294:
2289:
2284:
2279:
2270:
2265:
2259:
2258:
2252:
2243:
2238:
2233:
2227:
2218:
2213:
2207:
2206:
2200:
2191:
2185:
2180:
2174:
2167:
2161:
2137:
2128:
2119:
2110:
2104:
2103:
2071:
2065:
2063:
2060:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
1996:
1993:
1988:
1984:
1963:
1938:
1934:
1930:
1926:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1896:
1892:
1888:
1883:
1879:
1875:
1853:
1849:
1845:
1842:
1837:
1833:
1829:
1824:
1820:
1797:
1793:
1770:
1766:
1762:
1758:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1728:
1724:
1720:
1717:
1714:
1694:
1691:
1688:
1685:
1680:
1676:
1661:
1652:
1649:
1626:
1623:
1622:
1621:
1609:
1604:is the set of
1596:In this case:
1571:
1568:
1567:
1566:
1558:
1539:
1497:is the set of
1473:Mathematician
1468:zero-sum games
1453:
1450:
1442:
1439:
1388:
1387:
1286:
1283:
1277:
1266:
1261:
1260:
1247:
1242:
1239:
1236:
1233:
1230:
1227:
1225:
1221:
1217:
1214:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1193:
1189:
1185:
1182:
1181:
1179:
1174:
1171:
1168:
1165:
1162:
1147:
1144:
1136:
1135:
1122:
1117:
1114:
1111:
1108:
1105:
1102:
1100:
1096:
1092:
1089:
1088:
1085:
1082:
1079:
1076:
1074:
1071:
1067:
1063:
1060:
1057:
1053:
1049:
1046:
1043:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1022:
1018:
1014:
1011:
1010:
1008:
1003:
1000:
997:
994:
991:
953:
950:
938:
937:
924:
919:
916:
913:
910:
907:
904:
902:
898:
894:
891:
890:
887:
884:
881:
878:
876:
873:
870:
867:
864:
859:
858:
855:
852:
849:
846:
843:
840:
838:
834:
830:
827:
826:
824:
819:
816:
813:
810:
807:
795:The function:
792:
789:
772:
769:
765:
761:
758:
755:
749:
746:
742:
738:
735:
732:
729:
726:
723:
720:
717:
714:
705:The function:
702:
699:
685:
682:
681:
680:
645:
642:
630:
627:
624:
621:
618:
615:
595:
575:
570:
566:
562:
559:
556:
551:
547:
526:
523:
518:
514:
493:
490:
485:
481:
457:
453:
450:
446:
440:
436:
432:
409:
405:
402:
398:
392:
388:
384:
347:) |
329:
326:
323:correspondence
310:
307:
304:
234:
229:
226:
225:
224:
209:
208:
207:
186:
148:
110:
107:
46:. It provides
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4532:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4502:
4500:
4485:
4477:
4476:
4473:
4467:
4464:
4462:
4459:
4457:
4456:Weak topology
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4438:
4436:
4432:
4425:
4421:
4418:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4396:
4393:
4391:
4388:
4386:
4383:
4381:
4380:Index theorem
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4362:
4360:
4356:
4350:
4347:
4345:
4342:
4341:
4339:
4337:Open problems
4335:
4329:
4326:
4324:
4321:
4319:
4316:
4314:
4311:
4309:
4306:
4304:
4301:
4300:
4298:
4294:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4239:
4237:
4233:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4194:
4192:
4189:
4187:
4184:
4183:
4181:
4179:
4175:
4165:
4162:
4160:
4157:
4155:
4152:
4149:
4145:
4141:
4138:
4136:
4133:
4131:
4128:
4127:
4125:
4121:
4115:
4112:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4066:
4063:
4060:
4056:
4051:
4047:
4043:
4036:
4031:
4029:
4024:
4022:
4017:
4016:
4013:
4005:
4001:
4000:
3995:
3991:
3990:
3986:
3979:
3972:
3966:
3961:
3960:
3954:
3950:
3943:
3939:
3935:
3928:
3924:
3920:
3919:
3915:
3907:
3901:
3897:
3893:
3886:
3883:
3878:
3872:
3865:
3864:
3859:
3853:
3851:
3849:
3845:
3840:
3836:
3831:
3826:
3821:
3816:
3812:
3808:
3804:
3800:
3796:
3789:
3786:
3781:
3777:
3773:
3769:
3765:
3761:
3757:
3753:
3746:
3743:
3738:
3734:
3730:
3724:
3720:
3713:
3710:
3705:
3701:
3697:
3693:
3689:
3685:
3684:
3676:
3673:
3668:
3662:
3658:
3657:
3652:
3646:
3643:
3638:
3631:
3629:
3625:
3620:
3616:
3609:
3606:
3601:
3599:0-521-38808-2
3595:
3591:
3584:
3581:
3576:
3572:
3567:
3562:
3557:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3522:
3520:
3516:
3511:
3507:
3503:
3499:
3495:
3489:
3487:
3483:
3477:
3475:
3472:
3464:
3460:
3458:
3454:
3450:
3447:
3443:
3440:
3436:
3432:
3427:
3426:
3425:
3423:
3418:
3416:
3409:
3407:
3404:
3400:
3397:
3393:
3389:
3384:
3383:
3382:
3376:
3373: ∈
3372:
3368:
3364:
3360:
3356:
3352:
3348:
3344:
3340:
3337:
3334:
3331:} is open in
3330:
3326:
3322:
3318:
3315: ⊂
3314:
3311:
3308:if for every
3307:
3306:
3301:
3297:
3294:
3293:
3292:
3290:
3286:
3282:
3274:
3272:
3270:
3266:
3262:
3258:
3251:
3247:
3245:
3239:
3235:
3234:combinatorial
3231:
3228:
3224:
3223:
3222:
3220:
3216:
3214:
3205:
3201:
3198:
3196:
3194:
3190:
3186:
3182:
3178:
3157:
3153:
3148:
3139:
3135:
3131:
3126:
3122:
3114:
3110:
3106:
3103:
3097:
3094:
3090:
3086:
3081:
3077:
3072:
3064:
3060:
3056:
3051:
3047:
3039:
3035:
3031:
3028:
3022:
3018:
3015:
3008:
3007:
3006:
3004:
3000:
2994:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2964:
2957:
2954:
2953:
2949:
2945:
2941:
2937:
2933:
2929:
2928:
2927:
2924:
2922:
2918:
2914:
2910:
2906:
2897:
2893:
2888:
2884:
2879:
2875:
2870:
2866:
2862:
2858:
2855:
2854:
2853:
2850:
2846:
2841:
2837:
2832:
2830:
2826:
2822:
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2782:
2778:
2775:
2770:
2766:
2761:
2757:
2752:
2748:
2743:
2739:
2735:
2731:
2727:
2722:
2718:
2713:
2709:
2704:
2700:
2695:
2691:
2687:
2680:
2677:
2676:
2675:
2671:
2667:
2661:
2657:
2651:
2647:
2641:
2637:
2629:
2623:
2619:
2616:
2613:
2609:
2603:
2599:
2596:
2594:
2588:
2584:
2581:
2578:
2574:
2568:
2564:
2561:
2560:
2559:
2557:
2553:
2550:) such that
2549:
2545:
2541:
2533:
2529:
2523:
2519:
2516:
2514:
2508:
2504:
2501:
2498:
2494:
2488:
2484:
2481:
2479:
2473:
2469:
2466:
2465:
2464:
2462:
2458:
2454:
2450:
2445:
2443:
2439:
2430:
2426:
2421:
2417:
2413:
2410:
2409:
2408:
2405:
2401:
2396:
2392:
2387:
2383:
2378:
2374:
2369:
2364:
2357:
2350:
2343:
2336:
2329:
2322:
2317:
2306:
2302:
2297:
2293:
2290:
2288:
2285:
2282:
2278:
2273:
2269:
2266:
2264:
2261:
2260:
2255:
2251:
2246:
2242:
2239:
2237:
2234:
2230:
2226:
2221:
2217:
2214:
2212:
2209:
2208:
2203:
2199:
2194:
2190:
2184:
2181:
2177:
2173:
2166:
2160:
2157:
2156:
2153:
2152:
2151:
2149:
2145:
2140:
2136:
2131:
2127:
2122:
2118:
2113:
2109:
2102:
2099:
2098:
2097:
2095:
2090:
2088:
2078:
2069:
2066:
2062:Proof outline
2061:
2058:
2041:
2035:
2032:
2029:
2023:
2020:
2017:
1994:
1986:
1982:
1961:
1952:
1936:
1932:
1928:
1917:
1911:
1908:
1902:
1894:
1890:
1886:
1881:
1877:
1851:
1847:
1843:
1835:
1831:
1822:
1818:
1795:
1791:
1768:
1764:
1760:
1749:
1743:
1740:
1734:
1726:
1722:
1715:
1712:
1692:
1686:
1683:
1678:
1674:
1660:
1658:
1648:
1646:
1642:
1638:
1632:
1625:Fair division
1624:
1618:
1614:
1610:
1607:
1603:
1600:The base set
1599:
1598:
1597:
1594:
1592:
1588:
1583:
1577:
1569:
1563:
1559:
1556:
1552:
1548:
1544:
1540:
1538:
1535:
1534:
1529:
1525:
1522:
1521:
1516:
1512:
1508:
1504:
1500:
1496:
1493:The base set
1492:
1491:
1490:
1488:
1484:
1480:
1476:
1471:
1469:
1465:
1459:
1451:
1448:
1440:
1438:
1436:
1432:
1431:metric spaces
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1391:
1386:
1384:
1379:
1376:
1372:
1369: ∈
1368:
1365:
1361:
1357:
1354:
1351:
1348:
1346:
1343:
1337:
1333:
1330:
1326:
1323:
1322:
1318:
1315:
1311:
1307:
1302:
1299:
1296:
1295:
1294:
1292:
1284:
1282:
1280:
1273:
1269:
1240:
1237:
1234:
1231:
1228:
1223:
1219:
1215:
1208:
1205:
1202:
1199:
1196:
1191:
1187:
1183:
1177:
1172:
1166:
1160:
1153:
1152:
1151:
1145:
1143:
1141:
1115:
1112:
1109:
1106:
1103:
1098:
1094:
1090:
1083:
1080:
1077:
1069:
1065:
1061:
1058:
1055:
1051:
1047:
1037:
1034:
1031:
1028:
1025:
1020:
1016:
1012:
1006:
1001:
995:
989:
982:
981:
980:
977:
975:
971:
967:
958:
951:
949:
947:
943:
917:
914:
911:
908:
905:
900:
896:
892:
885:
882:
879:
871:
868:
865:
853:
850:
847:
844:
841:
836:
832:
828:
822:
817:
811:
805:
798:
797:
796:
790:
788:
786:
767:
763:
759:
756:
753:
747:
744:
740:
736:
733:
730:
724:
718:
712:
700:
695:
690:
683:
678:
674:
671: ∈
670:
666:
662:
658:
655: ∈
654:
650:
646:
643:
625:
619:
616:
613:
593:
568:
564:
557:
554:
549:
545:
524:
516:
512:
491:
483:
479:
451:
448:
438:
434:
403:
400:
390:
386:
374:
370:
367: ×
366:
362:
358:
354:
351: ∈
350:
346:
342:
339:if the set {(
338:
334:
330:
327:
324:
308:
305:
302:
294:
290:
286:
282:
279:, written as
278:
274:
270:
266:
262:
258:
254:
250:
246:
243:from the set
242:
239:
235:
232:
231:
227:
223:
221:
216:
213:
210:
205:
202: ∈
201:
198:
194:
190:
187:
185:
182:
178:
175:
174:
173:
170:
167:
165:
159:
155:
152:
149:
146:
143:
142:
138:
135:
131:
127:
122:
119:
116:
115:
114:
108:
106:
104:
100:
96:
92:
88:
83:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4446:Balanced set
4420:Distribution
4358:Applications
4211:Krein–Milman
4196:Closed graph
3997:
3958:
3941:
3926:
3895:
3885:
3869:. Springer.
3862:
3802:
3798:
3788:
3780:the original
3759:
3755:
3745:
3718:
3712:
3687:
3683:Econometrica
3681:
3675:
3655:
3645:
3636:
3618:
3608:
3589:
3583:
3541:(1): 48–49.
3538:
3534:
3501:
3497:
3468:
3457:fixed points
3428:
3419:
3412:
3385:
3380:
3374:
3370:
3366:
3363:Kakutani map
3362:
3358:
3354:
3353:and φ:
3346:
3342:
3338:Kakutani map
3332:
3328:
3324:
3320:
3316:
3312:
3303:
3299:
3278:
3264:
3260:
3256:
3254:
3249:
3243:
3218:
3212:
3209:
3203:
3199:
3192:
3188:
3184:
3180:
3176:
3174:
2998:
2995:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2962:
2960:
2955:
2947:
2943:
2939:
2935:
2931:
2925:
2920:
2916:
2912:
2908:
2904:
2902:
2895:
2891:
2886:
2882:
2877:
2873:
2868:
2864:
2860:
2856:
2848:
2844:
2839:
2835:
2833:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2768:
2764:
2759:
2755:
2750:
2746:
2741:
2737:
2720:
2716:
2711:
2707:
2702:
2698:
2693:
2689:
2683:
2678:
2669:
2665:
2659:
2655:
2649:
2645:
2639:
2635:
2633:
2627:
2621:
2617:
2611:
2607:
2601:
2597:
2592:
2586:
2582:
2576:
2572:
2566:
2562:
2555:
2551:
2547:
2543:
2539:
2537:
2531:
2527:
2521:
2517:
2512:
2506:
2502:
2496:
2492:
2486:
2482:
2477:
2471:
2467:
2460:
2456:
2455:) such that
2452:
2448:
2446:
2437:
2435:
2428:
2424:
2419:
2415:
2411:
2403:
2399:
2394:
2390:
2385:
2381:
2376:
2372:
2370:
2362:
2355:
2348:
2341:
2334:
2327:
2320:
2318:
2314:
2304:
2300:
2295:
2291:
2286:
2280:
2276:
2271:
2267:
2262:
2253:
2249:
2244:
2240:
2235:
2228:
2224:
2219:
2215:
2210:
2201:
2197:
2192:
2188:
2182:
2175:
2171:
2164:
2158:
2143:
2138:
2134:
2129:
2125:
2120:
2116:
2111:
2107:
2105:
2100:
2091:
2084:
2067:
1953:
1665:
1654:
1634:
1616:
1612:
1601:
1595:
1579:
1554:
1550:
1546:
1542:
1536:
1532:
1531:
1527:
1523:
1519:
1518:
1510:
1506:
1494:
1472:
1461:
1441:Applications
1434:
1425:. Since all
1422:
1418:
1414:
1410:
1406:
1402:
1401:range space
1392:
1389:
1380:
1377:
1374:
1370:
1366:
1363:
1359:
1355:
1352:
1349:
1339:
1335:
1331:
1328:
1324:
1303:
1300:
1297:
1288:
1275:
1271:
1264:
1262:
1149:
1139:
1137:
978:
973:
969:
965:
963:
945:
941:
939:
794:
784:
704:
693:
676:
672:
668:
664:
660:
656:
652:
648:
647:Let φ:
368:
364:
356:
352:
348:
344:
340:
337:closed graph
336:
332:
328:Closed graph
322:
292:
288:
284:
280:
276:
268:
260:
256:
252:
248:
244:
240:
237:
217:
214:
211:
203:
199:
196:
192:
188:
184:
181:closed graph
176:
171:
168:
161:
157:
153:
150:
144:
123:
120:
117:
112:
84:
58:subset of a
35:
29:
4375:Heat kernel
4365:Hardy space
4272:Trace class
4186:Hahn–Banach
4148:Topological
3981:economics.)
3944:. Springer.
3471:Ken Binmore
3429:Let S be a
3386:Let S be a
3319:, the set {
2834:But since (
2777:subsequence
2730:compact set
1479:game theory
1458:Game theory
1452:Game theory
1383:fixed point
661:fixed point
644:Fixed point
247:to the set
228:Definitions
220:fixed point
99:game theory
64:fixed point
4499:Categories
4308:C*-algebra
4123:Properties
3478:References
3248:Arbitrary
3215:-simplices
2774:convergent
1810:such that
1705:such that
1629:See also:
1574:See also:
1456:See also:
1445:See also:
1270:= 0.5 - 1/
606:, we have
471:such that
363:subset of
255:points in
62:to have a
4282:Unbounded
4277:Transpose
4235:Operators
4164:Separable
4159:Reflexive
4144:Algebraic
4130:Barrelled
4004:EMS Press
3737:984777840
3431:non-empty
3422:Euclidean
3403:Hausdorff
3388:non-empty
3323:| φ(
3158:∗
3140:∗
3132:−
3127:∗
3115:∗
3107:−
3098:−
3082:∗
3065:∗
3057:−
3052:∗
3040:∗
3032:−
2981:*. Since
2907:* equals
2881:) = lim (
2872:) − (lim
2863:* = (lim
2333:= 1. Let
2042:φ
2036:
2030:∈
1992:→
1918:φ
1912:
1903:∈
1750:φ
1744:
1735:⊂
1716:
1690:→
1637:envy-free
1560:Then the
1475:John Nash
1399:Hausdorff
1306:non-empty
1238:≤
1232:≤
1200:≤
1161:φ
1113:≤
1029:≤
990:φ
915:≤
845:≤
806:φ
757:−
734:−
713:φ
620:ϕ
617:∈
558:ϕ
555:∈
522:→
489:→
452:∈
404:∈
306:∈
273:power set
126:non-empty
109:Statement
103:economics
91:John Nash
4484:Category
4296:Algebras
4178:Theorems
4135:Complete
4104:Schwartz
4050:glossary
3925:(1989).
3839:16589065
3653:(1997).
3617:(1994).
3575:16588946
3529:(1950).
3465:Anecdote
3310:open set
3206:-simplex
3183:) since
2326:= 0 and
2148:sequence
1319:of some
684:Examples
586:for all
359:)} is a
265:function
139:of some
76:topology
4287:Unitary
4267:Nuclear
4252:Compact
4247:Bounded
4242:Adjoint
4216:Min–max
4109:Sobolev
4094:Nuclear
4084:Hilbert
4079:Fréchet
4044: (
4006:, 2001
3830:1063516
3807:Bibcode
3776:2032478
3704:1907539
3566:1063129
3543:Bibcode
3435:compact
3392:compact
2969:* then
2911:*. Let
2807:*) and
2007:. Then
1565:exists.
1409::
1334::
1310:compact
1281:= 3/4.
1142:= 0.5.
371:in the
283::
271:to the
253:or more
156::
130:compact
56:compact
4262:Normal
4099:Orlicz
4089:Hölder
4069:Banach
4058:Spaces
4046:topics
3967:
3902:
3873:
3837:
3827:
3774:
3735:
3725:
3702:
3663:
3596:
3573:
3563:
3444:of a
3442:subset
3439:convex
3424:case:
3399:subset
3396:convex
3302:→2 is
3289:Ky Fan
3187:* and
3003:convex
2985:* ∈ φ(
2946:* ∈ φ(
2899:) = 0.
2811:* ∈ φ(
2803:* ∈ φ(
2442:convex
2205:) ≤ 2
2142:) for
2074:": -->
1954:Since
1606:tuples
1587:Walras
1526:Then,
1499:tuples
1433:) and
1381:has a
1340:be an
1317:subset
1314:convex
751:
361:closed
218:has a
177:φ has
137:subset
134:convex
68:mapped
52:convex
34:, the
4074:Besov
3772:JSTOR
3700:JSTOR
3401:of a
3202:is a
3001:) is
2436:Then
2170:>
2106:Let (
2033:graph
1909:graph
1866:, so
1741:graph
1713:graph
1663:Proof
1304:be a
659:is a
267:from
162:be a
124:be a
38:is a
4422:(or
4140:Dual
3965:ISBN
3900:ISBN
3871:ISBN
3835:PMID
3733:OCLC
3723:ISBN
3661:ISBN
3594:ISBN
3571:PMID
3437:and
3394:and
3345:and
3341:Let
3287:and
3005:and
2973:* =
2965:* =
2938:* ≤
2934:) ∋
2919:* =
2903:So,
2859:* −
2827:* ≤
2819:* ≥
2664:and
2546:∈ φ(
2451:∈ φ(
2432:)/2.
2398:and
2248:∈ φ(
2223:∈ φ(
2179:≥ 0
2163:1 ≥
2076:edit
1639:and
1553:, φ(
1375:Then
1312:and
1206:<
1107:<
1035:<
909:<
851:<
696:(x)=
537:and
423:and
212:Then
132:and
101:and
42:for
3825:PMC
3815:doi
3764:doi
3692:doi
3561:PMC
3551:doi
3506:doi
3349:be
3283:by
2989:),
2961:If
2923:*.
2831:*.
2787:*,
2779:by
2732:by
2414:= (
1580:In
1517:in
1501:of
1338:→2
1329:Let
1298:Let
1274:,
1229:0.5
1209:0.5
1104:0.5
1084:0.5
1038:0.5
906:0.5
886:0.5
854:0.5
667:if
663:of
275:of
151:Let
118:Let
30:In
4501::
4048:–
4002:,
3996:,
3894:.
3847:^
3833:.
3823:.
3813:.
3803:38
3801:.
3797:.
3770:.
3758:.
3754:.
3731:.
3698:.
3688:22
3686:.
3627:^
3569:.
3559:.
3549:.
3539:36
3537:.
3533:.
3518:^
3500:.
3485:^
3433:,
3417:.
3390:,
2977:=
2950:).
2942:≤
2930:φ(
2915:=
2890:−
2843:−
2795:*,
2791:*,
2763:,
2754:,
2745:,
2715:,
2706:,
2697:,
2672:+1
2662:+1
2654:,
2652:+1
2644:,
2642:+1
2626:=
2624:+1
2606:=
2604:+1
2591:=
2589:+1
2571:=
2569:+1
2554:≤
2526:=
2524:+1
2511:=
2509:+1
2491:=
2489:+1
2476:=
2474:+1
2459:≥
2444:.
2389:,
2380:,
2368:.
2299:≤
2287:6.
2275:≥
2263:5.
2257:)
2236:4.
2232:)
2211:3.
2196:−
2183:2.
2159:1.
2133:,
2124:,
2115:,
2070:=
1951:.
1659::
1647:.
1593:.
1373:.
1362:)
1347:on
1327:.
1308:,
679:).
504:,
236:A
195:)
179:a
166:on
128:,
105:.
54:,
4426:)
4150:)
4146:/
4142:(
4052:)
4034:e
4027:t
4020:v
3973:.
3908:.
3879:.
3841:.
3817::
3809::
3766::
3760:3
3739:.
3706:.
3694::
3669:.
3602:.
3577:.
3553::
3545::
3512:.
3508::
3502:8
3377:.
3375:X
3371:x
3367:x
3359:Y
3355:X
3347:Y
3343:X
3335:.
3333:X
3329:W
3325:x
3321:x
3317:Y
3313:W
3300:X
3265:S
3261:S
3257:S
3250:S
3219:n
3213:n
3204:n
3200:S
3193:x
3189:q
3185:p
3181:x
3177:x
3154:q
3149:)
3136:q
3123:p
3111:q
3104:x
3095:1
3091:(
3087:+
3078:p
3073:)
3061:q
3048:p
3036:q
3029:x
3023:(
3019:=
3016:x
2999:x
2991:x
2987:x
2983:p
2979:q
2975:x
2971:p
2967:q
2963:p
2948:x
2944:p
2940:x
2936:q
2932:x
2921:a
2917:b
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1983:x
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