529:
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1564:
1471:
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2125:
1638:
1606:
1427:
1387:
1309:
1971:
954:
2181:
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1345:
1266:
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2925:
2071:
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2718:
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1939:
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2742:
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1149:
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230:
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2800:
2491:
2393:
Kolmogorov, A. N., & Fomin, S. V. (1967). Elements of the theory of functions and functional analysis. Courier Dover
Publications.
185:
3827:
2457:
1055:
576:
2856:
2396:
Stein, Elias; Shakarchi, R. (2011). Functional
Analysis: An Introduction to Further Topics in Analysis. Princeton University Press.
3817:
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550:
58:
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416:
215:
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3156:
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2653:
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3706:
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2770:
1200:
of finite dimension. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets
3731:
3711:
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1203:
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490:
170:
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2010:
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3637:
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3235:
3144:
2826:
2683:
1796:
1480:
250:
3775:
2889:
2365:{\displaystyle \|y\|_{\infty }\equiv \max _{a\leq x\leq b}|y(x)|\qquad {\text{where}}\ \ y\in {\mathcal {C}}(a,b)}
1750:
3680:
3456:
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2821:
1142:
338:
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3936:
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1531:
1438:
1193:
976:
539:
3161:
2831:
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3617:
3375:
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3028:
2518:
558:
543:
433:
200:
125:
80:
3900:
3800:
3622:
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3190:
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2426:
1833:
1115:
478:
466:
33:
2187:
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1611:
1573:
1394:
1354:
1276:
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3409:
3404:
3246:
3129:
3087:
2673:
2565:
1945:
1524:
1312:
1127:
1039:
933:
396:
3770:
3736:
3644:
3354:
3309:
3151:
3074:
2851:
2737:
2622:
2153:
2048:
1840:
1319:
1268:
1240:
774:{\displaystyle {\begin{aligned}(f+g)(x)&=f(x)+g(x)\\(c\cdot f)(x)&=c\cdot f(x)\end{aligned}}}
470:
381:
343:
155:
66:
3753:
3743:
3589:
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3108:
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2795:
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2617:
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2128:
1645:
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972:
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315:
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3905:
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3208:
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2148:
2002:
1787:
1177:
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1092:
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1062:
426:
310:
285:
3822:
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3519:
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3124:
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2765:
2487:
2453:
1566:
1028:
498:
462:
386:
290:
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265:
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1873:
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3524:
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2967:
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2727:
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2411:
1790:
1520:
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1347:
1047:
376:
353:
110:
1917:
497:
addition and scalar multiplication. In other scenarios, the function space might inherit a
3915:
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3178:
2421:
2132:
1907:
1119:
1104:
1088:
1066:
790:
371:
348:
295:
1993:
481:
which is inherited by the function space. For example, the set of functions from any set
3805:
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3324:
3304:
3264:
3254:
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2596:
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2224:
1783:
1008:
999:
1038:, with utility depending on the nature of the spaces. A commonly used example is the
3930:
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1903:
1138:
1134:
1123:
984:
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on the space of set theoretic functions (i.e. not necessarily continuous functions)
3504:
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3166:
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1197:
988:
518:
502:
486:
95:
3203:
1998:
2447:
3369:
1988:
1027:, one may attempt to put a topology on the space of continuous functions from a
841:
528:
454:
17:
2486:(Unabridged repr. ed.). Mineola, New York: Dover Publications. p. 6.
3213:
3002:
2693:
1043:
875:
833:
806:
275:
270:
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in the above, and many of the major examples are function spaces carrying a
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793:) of all such functions which respect that structure. For example, if
1738:{\textstyle \|f\|_{p}=\left(\int _{\mathbb {R} }|f|^{p}\right)^{1/p}}
786:
1192:
is organized around adequate techniques to bring function spaces as
3020:
858:
of a function space with no extra structure can be found by the
3024:
2507:
1069:
is essentially that of discrete invariants of function spaces;
522:
1054:. In this context, this topology is also referred to as the
613:
where the operations are defined pointwise, that is, for any
2503:
2342:
2193:
2097:
1952:
1803:
1756:
187:
1141:
that can model lambda calculus, by creating a well-behaved
851:
with addition and scalar multiplication defined pointwise.
785:
has additional structure, one might consider instead the
870:
Function spaces appear in various areas of mathematics:
1658:
1103:; but as (single) functor, of type , it appears as an
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2258:
2190:
2156:
2094:
2057:
2013:
1948:
1920:
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1397:
1357:
1322:
1279:
1243:
1206:
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909:
may be identified with the set of all functions from
651:
2452:. Springer Science & Business Media. p. 4.
1076:, the basic technical problem is how to construct a
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1227:
948:
773:
609:can be given the structure of a vector space over
1228:{\displaystyle \Omega \subseteq \mathbb {R} ^{n}}
112:
2279:
1429:continuous functions that have continuous first
232:
1152:, given two finite-dimensional representations
1137:, the basic idea is to find constructions from
960:! may be used for permutations of a single set
2040:{\displaystyle {\text{Lip}}_{0}(\mathbb {R} )}
1196:within reach of the ideas that would apply to
1099:. It appears in one way as the representation
3036:
2519:
1985:all functions, space of pointwise convergence
1826:{\displaystyle {\mathcal {S}}'(\mathbb {R} )}
1513:{\displaystyle C_{c}^{\infty }(\mathbb {R} )}
1389:continuous functions which vanish at infinity
434:
8:
3781:RieszâMarkovâKakutani representation theorem
2266:
2259:
2164:
2157:
1775:{\displaystyle {\mathcal {S}}(\mathbb {R} )}
1666:
1659:
88:
1982:continuous functions, compact open topology
557:. Unsourced material may be challenged and
3876:Vitale's random BrunnâMinkowski inequality
3793:
3043:
3029:
3021:
2943:Vitale's random BrunnâMinkowski inequality
2900:
2526:
2512:
2504:
441:
427:
97:
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2012:
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1613:
1591:
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1581:
1575:
1559:{\displaystyle C^{\omega }(\mathbb {R} )}
1549:
1548:
1539:
1533:
1503:
1502:
1493:
1488:
1482:
1466:{\displaystyle C^{\infty }(\mathbb {R} )}
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1219:
1215:
1214:
1205:
1168:over the vector space of linear maps Hom(
1012:. It consists of the set of all possible
935:
820:with pointwise operations (often denoted
652:
650:
577:Learn how and when to remove this message
2438:
415:
362:
329:
249:
217:
208:
64:
57:
32:
27:Set of functions between two fixed sets
1271:endowed with the uniform norm topology
1150:representation theory of finite groups
2480:(2000). Silverman, Richard A. (ed.).
2449:Representation Theory: A First Course
2446:Fulton, William; Harris, Joe (1991).
236:
221:
202:
191:
176:
157:
146:
127:
101:
82:
7:
3889:Applications & related
2956:Applications & related
2088:is an element of the function space
998:, the set of all functions from the
555:adding citations to reliable sources
172:
163:
2875:Marcinkiewicz interpolation theorem
2216:{\displaystyle {\mathcal {C}}(a,b)}
2120:{\displaystyle {\mathcal {C}}(a,b)}
1633:{\displaystyle 1\leq p\leq \infty }
1601:{\displaystyle L^{p}(\mathbb {R} )}
1422:{\displaystyle C^{r}(\mathbb {R} )}
1382:{\displaystyle C_{0}(\mathbb {R} )}
1304:{\displaystyle C_{c}(\mathbb {R} )}
1164:, one can form a representation of
519:Vector space § Function spaces
505:structure, hence the name function
469:between two fixed sets. Often, the
2801:Symmetric decreasing rearrangement
2705:
2270:
2168:
1966:{\displaystyle {\mathcal {O}}_{U}}
1627:
1494:
1447:
1207:
1091:, the function space is called an
983:; the best known examples include
975:linear transformations, including
949:{\displaystyle X\leftrightarrow Y}
25:
1868:compact support in limit topology
1056:topology of pointwise convergence
3818:Lebesgue differentiation theorem
3699:Carathéodory's extension theorem
1126:are used to express the idea of
527:
493:vector space structure given by
142:
133:
2322:
2176:{\displaystyle \|y\|_{\infty }}
1861:{\displaystyle D(\mathbb {R} )}
1340:{\displaystyle B(\mathbb {R} )}
1261:{\displaystyle C(\mathbb {R} )}
977:topologies on the vector spaces
59:History of the function concept
2407:List of mathematical functions
2359:
2347:
2318:
2314:
2308:
2301:
2210:
2198:
2114:
2102:
2034:
2026:
1855:
1847:
1820:
1812:
1769:
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1706:
1697:
1595:
1587:
1553:
1545:
1507:
1499:
1460:
1452:
1416:
1408:
1376:
1368:
1334:
1326:
1298:
1290:
1255:
1247:
940:
801:itself are vector spaces over
764:
758:
739:
733:
730:
718:
711:
705:
696:
690:
677:
671:
668:
656:
1:
2771:Convergence almost everywhere
2001:functions, also known as the
1107:to a functor of type (-×
2066:{\displaystyle \mathbb {R} }
878:, the set of functions from
3942:Topology of function spaces
3871:PrĂ©kopaâLeindler inequality
2938:PrĂ©kopaâLeindler inequality
2791:Locally integrable function
2713:{\displaystyle L^{\infty }}
3963:
3813:Lebesgue's density theorem
2684:Square-integrable function
1979:piecewise linear functions
1311:continuous functions with
956:. The factorial notation
832:)). One such space is the
601:be any set. The functions
516:
417:List of specific functions
3866:MinkowskiâSteiner formula
3796:
3681:Projection-valued measure
2933:MinkowskiâSteiner formula
1793:and its continuous dual,
1194:topological vector spaces
1143:Cartesian closed category
1046:. Also available is the
816:form a vector space over
3849:Isoperimetric inequality
3828:VitaliâHahnâSaks theorem
3157:Carathéodory's criterion
2916:Isoperimetric inequality
3854:BrunnâMinkowski theorem
3723:Decomposition theorems
2921:BrunnâMinkowski theorem
1896:{\displaystyle W^{k,p}}
1567:real analytic functions
1080:on a function space of
971:, the same is seen for
901:As a special case, the
860:ErdĆsâKaplansky theorem
3901:Descriptive set theory
3801:Disintegration theorem
3236:Universally measurable
2776:Convergence in measure
2714:
2483:Calculus of variations
2427:Functional determinant
2366:
2217:
2177:
2131:that are defined on a
2121:
2067:
2041:
1967:
1935:
1897:
1862:
1834:tempered distributions
1827:
1776:
1739:
1634:
1602:
1560:
1514:
1467:
1423:
1383:
1341:
1305:
1262:
1229:
1128:higher-order functions
1116:functional programming
950:
775:
3703:Convergence theorems
3162:Cylindrical Ï-algebra
2890:RieszâFischer theorem
2715:
2674:Polarization identity
2367:
2218:
2178:
2122:
2068:
2042:
1973:holomorphic functions
1968:
1936:
1934:{\displaystyle L^{p}}
1898:
1863:
1828:
1777:
1740:
1635:
1603:
1561:
1515:
1468:
1424:
1384:
1342:
1306:
1263:
1230:
1040:compact-open topology
951:
913:to {0, 1}, denoted 2.
776:
477:will have additional
3771:Minkowski inequality
3645:Cylinder set measure
3530:Infinite-dimensional
3145:equivalence relation
3075:Lebesgue integration
2895:RieszâThorin theorem
2738:Infimum and supremum
2697:
2623:Lebesgue integration
2256:
2188:
2154:
2129:continuous functions
2092:
2073:that vanish at zero.
2055:
2011:
1946:
1918:
1874:
1841:
1797:
1751:
1656:
1612:
1574:
1532:
1481:
1439:
1395:
1355:
1320:
1277:
1269:continuous functions
1241:
1204:
1084:(functions of time);
1082:paths of the process
1074:stochastic processes
934:
649:
551:improve this section
3766:Hölder's inequality
3628:of random variables
3590:Measurable function
3477:Particular measures
3066:Absolute continuity
2857:Young's convolution
2796:Measurable function
2679:Pythagorean theorem
2669:Parseval's identity
2618:Integrable function
2047:, the space of all
1906:of functions whose
1498:
1190:Functional analysis
1185:Functional analysis
1101:canonical bifunctor
1078:probability measure
996:functional analysis
969:functional analysis
3906:Probability theory
3231:Transverse measure
3209:Non-measurable set
3191:Locally measurable
2978:Probability theory
2880:Plancherel theorem
2786:Integral transform
2733:Chebyshev distance
2710:
2659:Euclidean distance
2592:Minkowski distance
2362:
2299:
2213:
2173:
2117:
2063:
2037:
1963:
1931:
1893:
1858:
1823:
1788:rapidly decreasing
1772:
1735:
1630:
1598:
1556:
1510:
1484:
1463:
1419:
1379:
1337:
1301:
1258:
1225:
1178:Hom representation
1093:exponential object
1063:algebraic topology
946:
842:linear functionals
771:
769:
251:Classes/properties
3924:
3923:
3884:
3883:
3613:almost everywhere
3559:Spherical measure
3457:Strictly positive
3385:Projection-valued
3125:Almost everywhere
3098:Probability space
3018:
3017:
2951:
2950:
2766:Almost everywhere
2551: &
2333:
2330:
2326:
2278:
2018:
1348:bounded functions
1072:In the theory of
1029:topological space
587:
586:
579:
513:In linear algebra
451:
450:
363:Generalizations
16:(Redirected from
3954:
3859:Milman's reverse
3842:
3840:Lebesgue measure
3794:
3198:
3184:infimum/supremum
3105:Measurable space
3045:
3038:
3031:
3022:
2968:Fourier analysis
2926:Milman's reverse
2909:
2907:Lebesgue measure
2901:
2885:RiemannâLebesgue
2728:Bounded function
2719:
2717:
2716:
2711:
2709:
2708:
2628:Taxicab geometry
2583:Measurable space
2528:
2521:
2514:
2505:
2498:
2497:
2470:
2464:
2463:
2443:
2412:Clifford algebra
2371:
2369:
2368:
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2345:
2331:
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2304:
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2222:
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2214:
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2196:
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2145:
2126:
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2118:
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2100:
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2069:
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2062:
2046:
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2043:
2038:
2033:
2025:
2024:
2019:
2016:
1976:linear functions
1972:
1970:
1969:
1964:
1962:
1961:
1956:
1955:
1940:
1938:
1937:
1932:
1930:
1929:
1908:weak derivatives
1902:
1900:
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1864:
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1811:
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1791:smooth functions
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1720:
1716:
1715:
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1700:
1695:
1694:
1693:
1674:
1673:
1648:functions whose
1639:
1637:
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1607:
1605:
1604:
1599:
1594:
1586:
1585:
1565:
1563:
1562:
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1552:
1544:
1543:
1521:smooth functions
1519:
1517:
1516:
1511:
1506:
1497:
1492:
1474:smooth functions
1472:
1470:
1469:
1464:
1459:
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1450:
1428:
1426:
1425:
1420:
1415:
1407:
1406:
1388:
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1375:
1367:
1366:
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1344:
1343:
1338:
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1310:
1308:
1307:
1302:
1297:
1289:
1288:
1267:
1265:
1264:
1259:
1254:
1234:
1232:
1231:
1226:
1224:
1223:
1218:
1048:product topology
955:
953:
952:
947:
886:may be denoted {
781:When the domain
780:
778:
777:
772:
770:
582:
575:
571:
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562:
531:
523:
443:
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429:
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226:
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121:
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114:
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105:
99:
91:
90:
86:
53:
30:
21:
18:Functional space
3962:
3961:
3957:
3956:
3955:
3953:
3952:
3951:
3937:Function spaces
3927:
3926:
3925:
3920:
3916:Spectral theory
3896:Convex analysis
3880:
3837:
3832:
3785:
3685:
3633:in distribution
3578:
3471:
3301:Logarithmically
3240:
3196:
3179:Essential range
3113:
3054:
3049:
3019:
3014:
2947:
2904:
2899:
2861:
2837:HausdorffâYoung
2817:BabenkoâBeckner
2805:
2754:
2700:
2695:
2694:
2688:
2632:
2601:
2597:Sequence spaces
2537:
2532:
2502:
2501:
2494:
2472:
2471:
2467:
2460:
2445:
2444:
2440:
2435:
2422:Spectral theory
2403:
2390:
2265:
2254:
2253:
2239:
2228:
2223:is the maximum
2186:
2185:
2163:
2152:
2151:
2135:
2133:closed interval
2090:
2089:
2083:
2080:
2053:
2052:
2014:
2009:
2008:
1949:
1944:
1943:
1921:
1916:
1915:
1877:
1872:
1871:
1839:
1838:
1800:
1795:
1794:
1749:
1748:
1704:
1684:
1683:
1679:
1678:
1665:
1654:
1653:
1610:
1609:
1577:
1572:
1571:
1535:
1530:
1529:
1525:compact support
1479:
1478:
1442:
1437:
1436:
1398:
1393:
1392:
1358:
1353:
1352:
1318:
1317:
1313:compact support
1280:
1275:
1274:
1239:
1238:
1213:
1202:
1201:
1187:
1175:
1171:
1167:
1163:
1159:
1155:
1120:lambda calculus
1105:adjoint functor
1089:category theory
1067:homotopy theory
1065:, the study of
1034:to another one
1016:of elements of
1000:natural numbers
932:
931:
868:
850:
846:
839:
831:
827:
819:
815:
811:
804:
800:
796:
784:
768:
767:
742:
715:
714:
680:
647:
646:
644:
640:
636:
632:
628:
624:
620:
616:
612:
608:
604:
600:
592:
583:
572:
566:
563:
548:
532:
521:
515:
484:
447:
411:
372:Binary relation
358:
325:
245:
239:
231:
224:
216:
205:
201:
194:
186:
179:
171:
160:
156:
149:
141:
130:
126:
119:
111:
104:
96:
85:
81:
40:
28:
23:
22:
15:
12:
11:
5:
3960:
3958:
3950:
3949:
3947:Linear algebra
3944:
3939:
3929:
3928:
3922:
3921:
3919:
3918:
3913:
3908:
3903:
3898:
3892:
3890:
3886:
3885:
3882:
3881:
3879:
3878:
3873:
3868:
3863:
3862:
3861:
3851:
3845:
3843:
3834:
3833:
3831:
3830:
3825:
3823:Sard's theorem
3820:
3815:
3810:
3809:
3808:
3806:Lifting theory
3797:
3791:
3787:
3786:
3784:
3783:
3778:
3773:
3768:
3763:
3762:
3761:
3759:FubiniâTonelli
3751:
3746:
3741:
3740:
3739:
3734:
3729:
3721:
3720:
3719:
3714:
3709:
3701:
3695:
3693:
3687:
3686:
3684:
3683:
3678:
3673:
3668:
3663:
3658:
3653:
3647:
3642:
3641:
3640:
3638:in probability
3635:
3625:
3620:
3615:
3609:
3608:
3607:
3602:
3597:
3586:
3584:
3580:
3579:
3577:
3576:
3571:
3566:
3561:
3556:
3551:
3550:
3549:
3539:
3534:
3533:
3532:
3522:
3517:
3512:
3507:
3502:
3497:
3492:
3487:
3481:
3479:
3473:
3472:
3470:
3469:
3464:
3459:
3454:
3449:
3444:
3439:
3434:
3429:
3424:
3419:
3418:
3417:
3412:
3407:
3397:
3392:
3387:
3382:
3372:
3367:
3362:
3357:
3352:
3347:
3345:Locally finite
3342:
3332:
3327:
3322:
3317:
3312:
3307:
3297:
3292:
3287:
3282:
3277:
3272:
3267:
3262:
3257:
3251:
3249:
3242:
3241:
3239:
3238:
3233:
3228:
3223:
3218:
3217:
3216:
3206:
3201:
3193:
3188:
3187:
3186:
3176:
3171:
3170:
3169:
3159:
3154:
3149:
3148:
3147:
3137:
3132:
3127:
3121:
3119:
3115:
3114:
3112:
3111:
3102:
3101:
3100:
3090:
3085:
3077:
3072:
3062:
3060:
3059:Basic concepts
3056:
3055:
3052:Measure theory
3050:
3048:
3047:
3040:
3033:
3025:
3016:
3015:
3013:
3012:
3011:
3010:
3005:
2995:
2990:
2985:
2980:
2975:
2970:
2965:
2959:
2957:
2953:
2952:
2949:
2948:
2946:
2945:
2940:
2935:
2930:
2929:
2928:
2918:
2912:
2910:
2898:
2897:
2892:
2887:
2882:
2877:
2871:
2869:
2863:
2862:
2860:
2859:
2854:
2849:
2844:
2839:
2834:
2829:
2824:
2819:
2813:
2811:
2807:
2806:
2804:
2803:
2798:
2793:
2788:
2783:
2781:Function space
2778:
2773:
2768:
2762:
2760:
2756:
2755:
2753:
2752:
2747:
2746:
2745:
2735:
2730:
2724:
2722:
2707:
2703:
2690:
2689:
2687:
2686:
2681:
2676:
2671:
2666:
2661:
2656:
2654:CauchyâSchwarz
2651:
2645:
2643:
2634:
2633:
2631:
2630:
2625:
2620:
2614:
2612:
2603:
2602:
2600:
2599:
2594:
2589:
2580:
2575:
2574:
2573:
2563:
2555:
2553:Hilbert spaces
2545:
2543:
2542:Basic concepts
2539:
2538:
2533:
2531:
2530:
2523:
2516:
2508:
2500:
2499:
2493:978-0486414485
2492:
2474:Gelfand, I. M.
2465:
2458:
2437:
2436:
2434:
2431:
2430:
2429:
2424:
2419:
2414:
2409:
2402:
2399:
2398:
2397:
2394:
2389:
2386:
2384:('sup norm').
2374:is called the
2361:
2358:
2355:
2352:
2349:
2344:
2339:
2336:
2320:
2316:
2313:
2310:
2307:
2303:
2297:
2294:
2291:
2288:
2285:
2281:
2277:
2272:
2268:
2264:
2261:
2225:absolute value
2212:
2209:
2206:
2203:
2200:
2195:
2170:
2166:
2162:
2159:
2116:
2113:
2110:
2107:
2104:
2099:
2079:
2076:
2075:
2074:
2061:
2036:
2032:
2028:
2023:
2006:
1996:
1991:
1986:
1983:
1980:
1977:
1974:
1960:
1954:
1941:
1928:
1924:
1890:
1887:
1884:
1880:
1869:
1857:
1853:
1849:
1846:
1836:
1822:
1818:
1814:
1810:
1805:
1784:Schwartz space
1771:
1767:
1763:
1758:
1746:
1732:
1728:
1724:
1719:
1713:
1708:
1703:
1699:
1692:
1687:
1682:
1677:
1672:
1668:
1664:
1661:
1629:
1626:
1623:
1620:
1617:
1597:
1593:
1589:
1584:
1580:
1569:
1555:
1551:
1547:
1542:
1538:
1527:
1509:
1505:
1501:
1496:
1491:
1487:
1476:
1462:
1458:
1454:
1449:
1445:
1434:
1418:
1414:
1410:
1405:
1401:
1390:
1378:
1374:
1370:
1365:
1361:
1350:
1336:
1332:
1328:
1325:
1315:
1300:
1296:
1292:
1287:
1283:
1272:
1257:
1253:
1249:
1246:
1222:
1217:
1212:
1209:
1186:
1183:
1182:
1181:
1173:
1169:
1165:
1161:
1157:
1153:
1146:
1139:partial orders
1131:
1124:function types
1112:
1085:
1070:
1059:
1021:
1009:sequence space
992:
985:Hilbert spaces
965:
945:
942:
939:
916:
915:
914:
867:
864:
848:
844:
837:
829:
825:
817:
813:
809:
802:
798:
794:
782:
766:
763:
760:
757:
754:
751:
748:
745:
743:
741:
738:
735:
732:
729:
726:
723:
720:
717:
716:
713:
710:
707:
704:
701:
698:
695:
692:
689:
686:
683:
681:
679:
676:
673:
670:
667:
664:
661:
658:
655:
654:
642:
638:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
590:
585:
584:
535:
533:
526:
514:
511:
482:
459:function space
449:
448:
446:
445:
438:
431:
423:
420:
419:
413:
412:
410:
409:
404:
399:
394:
389:
384:
379:
374:
368:
365:
364:
360:
359:
357:
356:
351:
346:
341:
335:
332:
331:
327:
326:
324:
323:
318:
313:
308:
303:
298:
293:
288:
283:
278:
273:
268:
263:
257:
254:
253:
247:
246:
244:
243:
237:
228:
222:
213:
203:
198:
192:
183:
177:
168:
158:
153:
147:
138:
128:
123:
117:
108:
102:
93:
83:
77:
74:
73:
62:
61:
55:
54:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3959:
3948:
3945:
3943:
3940:
3938:
3935:
3934:
3932:
3917:
3914:
3912:
3911:Real analysis
3909:
3907:
3904:
3902:
3899:
3897:
3894:
3893:
3891:
3887:
3877:
3874:
3872:
3869:
3867:
3864:
3860:
3857:
3856:
3855:
3852:
3850:
3847:
3846:
3844:
3841:
3835:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3807:
3804:
3803:
3802:
3799:
3798:
3795:
3792:
3790:Other results
3788:
3782:
3779:
3777:
3776:RadonâNikodym
3774:
3772:
3769:
3767:
3764:
3760:
3757:
3756:
3755:
3752:
3750:
3749:Fatou's lemma
3747:
3745:
3742:
3738:
3735:
3733:
3730:
3728:
3725:
3724:
3722:
3718:
3715:
3713:
3710:
3708:
3705:
3704:
3702:
3700:
3697:
3696:
3694:
3692:
3688:
3682:
3679:
3677:
3674:
3672:
3669:
3667:
3664:
3662:
3659:
3657:
3654:
3652:
3648:
3646:
3643:
3639:
3636:
3634:
3631:
3630:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:Convergence:
3610:
3606:
3603:
3601:
3598:
3596:
3593:
3592:
3591:
3588:
3587:
3585:
3581:
3575:
3572:
3570:
3567:
3565:
3562:
3560:
3557:
3555:
3552:
3548:
3545:
3544:
3543:
3540:
3538:
3535:
3531:
3528:
3527:
3526:
3523:
3521:
3518:
3516:
3513:
3511:
3508:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3482:
3480:
3478:
3474:
3468:
3465:
3463:
3460:
3458:
3455:
3453:
3450:
3448:
3445:
3443:
3440:
3438:
3435:
3433:
3430:
3428:
3425:
3423:
3420:
3416:
3415:Outer regular
3413:
3411:
3410:Inner regular
3408:
3406:
3405:Borel regular
3403:
3402:
3401:
3398:
3396:
3393:
3391:
3388:
3386:
3383:
3381:
3377:
3373:
3371:
3368:
3366:
3363:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3337:
3333:
3331:
3328:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3302:
3298:
3296:
3293:
3291:
3288:
3286:
3283:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3258:
3256:
3253:
3252:
3250:
3248:
3243:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3215:
3212:
3211:
3210:
3207:
3205:
3202:
3200:
3194:
3192:
3189:
3185:
3182:
3181:
3180:
3177:
3175:
3172:
3168:
3165:
3164:
3163:
3160:
3158:
3155:
3153:
3150:
3146:
3143:
3142:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3122:
3120:
3116:
3110:
3106:
3103:
3099:
3096:
3095:
3094:
3093:Measure space
3091:
3089:
3086:
3084:
3082:
3078:
3076:
3073:
3071:
3067:
3064:
3063:
3061:
3057:
3053:
3046:
3041:
3039:
3034:
3032:
3027:
3026:
3023:
3009:
3006:
3004:
3001:
3000:
2999:
2996:
2994:
2993:Sobolev space
2991:
2989:
2988:Real analysis
2986:
2984:
2981:
2979:
2976:
2974:
2973:Lorentz space
2971:
2969:
2966:
2964:
2963:Bochner space
2961:
2960:
2958:
2954:
2944:
2941:
2939:
2936:
2934:
2931:
2927:
2924:
2923:
2922:
2919:
2917:
2914:
2913:
2911:
2908:
2902:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2878:
2876:
2873:
2872:
2870:
2868:
2864:
2858:
2855:
2853:
2850:
2848:
2845:
2843:
2840:
2838:
2835:
2833:
2830:
2828:
2825:
2823:
2820:
2818:
2815:
2814:
2812:
2808:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2763:
2761:
2757:
2751:
2748:
2744:
2741:
2740:
2739:
2736:
2734:
2731:
2729:
2726:
2725:
2723:
2721:
2701:
2691:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2664:Hilbert space
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2646:
2644:
2642:
2640:
2635:
2629:
2626:
2624:
2621:
2619:
2616:
2615:
2613:
2611:
2609:
2604:
2598:
2595:
2593:
2590:
2588:
2584:
2581:
2579:
2578:Measure space
2576:
2572:
2569:
2568:
2567:
2564:
2562:
2560:
2556:
2554:
2550:
2547:
2546:
2544:
2540:
2536:
2529:
2524:
2522:
2517:
2515:
2510:
2509:
2506:
2495:
2489:
2485:
2484:
2479:
2475:
2469:
2466:
2461:
2459:9780387974958
2455:
2451:
2450:
2442:
2439:
2432:
2428:
2425:
2423:
2420:
2418:
2415:
2413:
2410:
2408:
2405:
2404:
2400:
2395:
2392:
2391:
2387:
2385:
2383:
2382:supremum norm
2379:
2378:
2372:
2356:
2353:
2350:
2337:
2334:
2311:
2305:
2295:
2292:
2289:
2286:
2283:
2275:
2262:
2250:
2246:
2242:
2235:
2231:
2226:
2207:
2204:
2201:
2183:
2160:
2150:
2143:
2139:
2134:
2130:
2111:
2108:
2105:
2086:
2077:
2051:functions on
2050:
2021:
2007:
2004:
2000:
1997:
1995:
1992:
1990:
1987:
1984:
1981:
1978:
1975:
1958:
1942:
1926:
1922:
1913:
1909:
1905:
1904:Sobolev space
1888:
1885:
1882:
1878:
1870:
1844:
1837:
1835:
1808:
1792:
1789:
1785:
1747:
1730:
1726:
1722:
1717:
1711:
1701:
1685:
1680:
1675:
1670:
1662:
1651:
1647:
1643:
1624:
1621:
1618:
1615:
1582:
1578:
1570:
1568:
1540:
1536:
1528:
1526:
1522:
1489:
1485:
1477:
1475:
1443:
1435:
1432:
1403:
1399:
1391:
1363:
1359:
1351:
1349:
1323:
1316:
1314:
1285:
1281:
1273:
1270:
1244:
1237:
1236:
1235:
1220:
1210:
1199:
1198:normed spaces
1195:
1191:
1184:
1179:
1176:) called the
1151:
1147:
1144:
1140:
1136:
1135:domain theory
1132:
1129:
1125:
1121:
1117:
1113:
1111:) on objects;
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1083:
1079:
1075:
1071:
1068:
1064:
1060:
1057:
1053:
1049:
1045:
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1033:
1030:
1026:
1022:
1019:
1015:
1011:
1010:
1005:
1001:
997:
993:
990:
989:Banach spaces
986:
982:
978:
974:
970:
966:
963:
959:
943:
937:
929:
925:
921:
917:
912:
908:
904:
900:
899:
897:
893:
889:
885:
881:
877:
873:
872:
871:
865:
863:
861:
857:
854:The cardinal
852:
843:
840:: the set of
835:
823:
808:
805:, the set of
792:
788:
761:
755:
752:
749:
746:
744:
736:
727:
724:
721:
708:
702:
699:
693:
687:
684:
682:
674:
665:
662:
659:
596:
581:
578:
570:
567:November 2017
560:
556:
552:
546:
545:
541:
536:This section
534:
530:
525:
524:
520:
512:
510:
508:
504:
500:
496:
492:
488:
480:
476:
472:
468:
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460:
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444:
439:
437:
432:
430:
425:
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422:
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408:
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403:
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398:
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388:
385:
383:
380:
378:
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370:
369:
367:
366:
361:
355:
352:
350:
347:
345:
342:
340:
337:
336:
334:
333:
330:Constructions
328:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
279:
277:
274:
272:
269:
267:
264:
262:
259:
258:
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242:
229:
227:
214:
212:
199:
197:
184:
182:
169:
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154:
152:
139:
137:
124:
122:
109:
107:
94:
92:
79:
78:
76:
75:
72:
68:
63:
60:
56:
51:
47:
43:
39:
38:
35:
31:
19:
3691:Main results
3427:Set function
3355:Metric outer
3310:Decomposable
3167:Cylinder set
3080:
2810:Inequalities
2780:
2750:Uniform norm
2638:
2607:
2558:
2482:
2478:Fomin, S. V.
2468:
2448:
2441:
2417:Tensor field
2388:Bibliography
2381:
2377:uniform norm
2375:
2373:
2248:
2244:
2240:
2233:
2229:
2147:
2141:
2137:
2084:
2081:
1994:Hölder space
1911:
1910:up to order
1649:
1433:derivatives.
1430:
1188:
1108:
1081:
1051:
1035:
1031:
1017:
1007:
1006:is called a
1003:
1002:to some set
961:
957:
927:
923:
910:
906:
895:
891:
887:
883:
879:
869:
853:
588:
573:
564:
549:Please help
537:
506:
487:vector space
458:
452:
397:Higher-order
391:
49:
45:
41:
3651:compact set
3618:of measures
3554:Pushforward
3547:Projections
3537:Logarithmic
3380:Probability
3370:Pre-measure
3152:Borel space
3070:of measures
3008:Von Neumann
2822:Chebyshev's
2184:defined on
1989:Hardy space
1160:of a group
930:is denoted
918:The set of
807:linear maps
499:topological
455:mathematics
382:Multivalued
344:Composition
339:Restriction
3931:Categories
3623:in measure
3350:Maximising
3320:Equivalent
3214:Vitali set
3003:C*-algebra
2827:Clarkson's
2433:References
1646:measurable
1097:map object
1044:loop space
973:continuous
920:bijections
876:set theory
834:dual space
637:, and any
517:See also:
316:Surjective
306:Measurable
301:Continuous
276:Polynomial
3737:Maharam's
3707:Dominated
3520:Intensity
3515:Hausdorff
3422:Saturated
3340:Invariant
3245:Types of
3204:Ï-algebra
3174:đ-system
3140:Borel set
3135:Baire set
2998:*-algebra
2983:Quasinorm
2852:Minkowski
2743:Essential
2706:∞
2535:Lp spaces
2338:∈
2293:≤
2287:≤
2276:≡
2271:∞
2267:‖
2260:‖
2169:∞
2165:‖
2158:‖
2049:Lipschitz
2003:Skorokhod
1745:is finite
1686:∫
1667:‖
1660:‖
1640:, is the
1628:∞
1625:≤
1619:≤
1541:ω
1495:∞
1448:∞
1211:⊆
1208:Ω
1014:sequences
941:↔
905:of a set
903:power set
856:dimension
797:and also
753:⋅
725:⋅
645:, define
538:does not
495:pointwise
479:structure
467:functions
321:Bijective
311:Injective
286:Algebraic
65:Types by
3754:Fubini's
3744:Egorov's
3712:Monotone
3671:variable
3649:Random:
3600:Strongly
3525:Lebesgue
3510:Harmonic
3500:Gaussian
3485:Counting
3452:Spectral
3447:Singular
3437:s-finite
3432:Ï-finite
3315:Discrete
3290:Complete
3247:Measures
3221:Null set
3109:function
2847:Markov's
2842:Hölder's
2832:Hanner's
2649:Bessel's
2587:function
2571:Lebesgue
2401:See also
1809:′
1025:topology
981:topology
866:Examples
791:subspace
621: :
597:and let
475:codomain
402:Morphism
387:Implicit
291:Analytic
281:Rational
266:Identity
261:Constant
71:codomain
48: (
34:Function
3666:process
3661:measure
3656:element
3595:Bochner
3569:Trivial
3564:Tangent
3542:Product
3400:Regular
3378:)
3365:Perfect
3338:)
3303:)
3295:Content
3285:Complex
3226:Support
3199:-system
3088:Measure
2867:Results
2566:Measure
2127:of all
1914:are in
1642:L space
1148:In the
1042:, e.g.
559:removed
544:sources
491:natural
485:into a
473:and/or
407:Functor
377:Partial
354:Inverse
3732:Jordan
3717:Vitali
3676:vector
3605:Weakly
3467:Vector
3442:Signed
3395:Random
3336:Quasi-
3325:Finite
3305:Convex
3265:Banach
3255:Atomic
3083:spaces
3068:
2720:spaces
2641:spaces
2610:spaces
2561:spaces
2549:Banach
2490:
2456:
2332:
2329:
2146:, the
1999:CĂ dlĂ g
1782:, the
1652:-norm
1608:, for
787:subset
629:, any
503:metric
489:has a
471:domain
296:Smooth
271:Linear
67:domain
3574:Young
3495:Euler
3490:Dirac
3462:Tight
3390:Radon
3360:Outer
3330:Inner
3280:Brown
3275:Borel
3270:Besov
3260:Baire
2325:where
2144:]
2136:[
2005:space
1523:with
922:from
894:} or
595:field
593:be a
507:space
461:is a
392:Space
3838:For
3727:Hahn
3583:Maps
3505:Haar
3376:Sub-
3130:Atom
3118:Sets
2905:For
2759:Maps
2488:ISBN
2454:ISBN
2238:for
2227:of
2149:norm
2078:Norm
1156:and
1118:and
987:and
789:(or
589:Let
542:any
540:cite
457:, a
69:and
2380:or
2280:max
2082:If
2017:Lip
1786:of
1644:of
1133:In
1114:In
1095:or
1087:In
1061:In
1023:In
994:In
967:In
926:to
882:to
874:In
836:of
822:Hom
641:in
633:in
553:by
501:or
465:of
463:set
453:In
3933::
2476:;
2252:,
2247:â€
2243:â€
2140:,
1122:,
898:.
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862:.
847:â
812:â
625:â
617:,
605:â
509:.
235:â
220:â
207:â
190:â
175:â
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115:â
113:đč
100:â
98:đč
89:đč
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3374:(
3334:(
3299:(
3197:Ï
3107:/
3081:L
3044:e
3037:t
3030:v
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2639:L
2608:L
2585:/
2559:L
2527:e
2520:t
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2496:.
2462:.
2360:)
2357:b
2354:,
2351:a
2348:(
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2335:y
2319:|
2315:)
2312:x
2309:(
2306:y
2302:|
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2208:b
2205:,
2202:a
2199:(
2194:C
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2112:b
2109:,
2106:a
2103:(
2098:C
2085:y
2060:R
2035:)
2031:R
2027:(
2022:0
1959:U
1953:O
1927:p
1923:L
1912:k
1889:p
1886:,
1883:k
1879:W
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1848:(
1845:D
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1817:R
1813:(
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1766:R
1762:(
1757:S
1731:p
1727:/
1723:1
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1712:p
1707:|
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1698:|
1691:R
1681:(
1676:=
1671:p
1663:f
1650:p
1622:p
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1592:R
1588:(
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1550:R
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1500:(
1490:c
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1453:(
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1413:R
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1180:.
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1170:V
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1162:G
1158:W
1154:V
1145:.
1130:.
1109:X
1058:.
1052:Y
1036:Y
1032:X
1020:.
1018:X
1004:X
991:.
964:.
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958:X
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888:X
884:Y
880:X
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838:X
830:V
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826:X
824:(
818:F
814:V
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783:X
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762:x
759:(
756:f
750:c
747:=
740:)
737:x
734:(
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728:f
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719:(
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709:x
706:(
703:g
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694:x
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