38:, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence. Below are some of the more common and typical examples. This article is intended as an introduction aimed to help practitioners explore appropriate techniques. The links below give details of necessary conditions and generalizations to more abstract settings. Proof techniques for the convergence of
2990:
1346:
If both above inequalities in the definition of a contraction mapping are weakened from "strictly less than" to "less than or equal to", the mapping is a non-expansion mapping. It is not sufficient to prove convergence to prove that
2582:
2875:-- In pointwise convergence, some (open) regions can converge arbitrarily slowly. With uniform convergence, there is a fixed convergence rate such that all points converge at least that fast. Formally,
1887:
141:
98:
2490:
2419:
2734:
2818:
942:
451:
2659:
3299:
3117:
2232:
364:
2311:
1556:
1064:
3215:
3186:
3085:
1835:
1708:
978:
774:
572:
244:
2542:
694:
185:
163:
1315:
505:
309:
2267:
397:
2096:
1194:
329:
1439:
1270:
3319:
3137:
2878:
2337:
2122:
1983:
1925:
1465:
1403:
2157:
3037:
1651:
1631:
1576:
1118:
1091:
478:
2867:
827:
641:
2578:
To consider the convergence of sequences of functions, it is necessary to define a distance between functions to replace the
Euclidean norm. These often include
3157:
3010:
2838:
2760:
2359:
2058:
2031:
2003:
1945:
1796:
1776:
1752:
1732:
1671:
1493:
1365:
1335:
1234:
1214:
1150:
1018:
998:
861:
798:
720:
612:
592:
282:
1611:
2560:
1714:. If these subsequences all have the same limit, then the original sequence also converges to that limit. If it can be shown that all of the subsequences of
1842:
This fact can be used directly and can also be used to prove the convergence of sequences that are not monotonic using techniques and theorems named for
3244:
3254:
3324:
3259:
3348:
1849:
1499:(the image of the domain), that is also sufficient for convergence. This also applies for decompositions. For example, consider
103:
60:
35:
2424:
2364:
2664:
1467:. However, the composition of a contraction mapping and a non-expansion mapping (or vice versa) is a contraction mapping.
2765:
870:
402:
2567:
2588:
3248:
3233:
2034:
1711:
3363:
For all of the above techniques, some form the basic analytic definition of convergence above applies. However,
3265:
3447:
3047:
3090:
2163:
261:
3408:
3352:
334:
2037:, but it can also be applied to sequences of iterates by replacing derivatives with discrete differences.
2556:
2272:
1502:
1023:
833:
3191:
3162:
3061:
1811:
1734:
must have the same limit, such as by showing that there is a unique fixed point of the transformation
1684:
3345:
establishing the pointwise (Lebesgue) almost everywhere convergence of
Fourier series of L2 functions
947:
725:
518:
202:
42:, a particular type of sequences corresponding to sums of many terms, are covered in the article on
3188:, such as the roll of a dice, and such a random variable is often spoken of informally as being in
2549:
2559:, a similar approach applies with Lyapunov functions replaced by Lyapunov functionals also called
2499:
646:
168:
146:
1275:
483:
287:
3217:, but convergence of sequence of random variables corresponds to convergence of the sequence of
2985:{\displaystyle \lim _{n\to \infty }\,\sup\{\,\left|f_{n}(x)-f_{\infty }(x)\right|:x\in A\,\}=0,}
2237:
369:
2066:
1155:
314:
258:
1408:
1239:
3342:
3337:
3304:
3122:
2739:
2316:
2101:
2010:
1950:
1892:
1444:
1370:
840:
777:
2127:
3015:
1636:
1616:
1561:
1096:
1069:
700:
456:
39:
2843:
803:
617:
8:
2872:
864:
28:
24:
20:
3330:
Each has its own proof techniques, which are beyond the current scope of this article.
3142:
2995:
2823:
2745:
2344:
2043:
2016:
1988:
1930:
1843:
1781:
1761:
1737:
1717:
1656:
1478:
1350:
1320:
1219:
1199:
1135:
1003:
983:
846:
783:
705:
597:
577:
267:
1581:
2736:. For this case, all of the above techniques can be applied with this function norm.
2006:
43:
1807:
3368:
3371:
space, it is possible for a sequence to converge to multiple different limits.
196:
188:
1020:. The composition of two contraction mappings is a contraction mapping, so if
3441:
1755:
515:
In many cases, the function whose convergence is of interest has the form
453:. The most direct proof technique from this definition is to find such a
3427:
697:
192:
2544:
can be found, although more complex forms are also common, for instance
2820:. For this case, the above techniques can be applied for each point
837:
1495:
is not a contraction mapping on its entire domain, but it is on its
3364:
1496:
800:
may be an elementwise operation, such as replacing each element of
2545:
1578:
is not a contraction mapping, but it is on the restricted domain
3367:
has its own definitions of convergence. For example, in a non-
3058:
Random variables are more complicated than simple elements of
2033:
is also convergent. Lyapunov's theorem is normally stated for
1798:, then the initial sequence must also converge to that limit.
507:
is not known in advance, the techniques below may be useful.
843:) then it is sufficient to prove convergence to prove that
1882:{\displaystyle V:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
1470:
832:
In such cases, if the problem satisfies the conditions of
136:{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} ^{n}}
93:{\displaystyle f:\mathbb {N} \rightarrow \mathbb {R} ^{n}}
2485:{\textstyle \lim _{k\rightarrow \infty }||f(k)||=\infty }
1128:
Famous examples of applications of this approach include
2573:
2496:
In many cases a quadratic
Lyapunov function of the form
2414:{\textstyle \lim _{k\rightarrow \infty }V(f(k))=\infty }
57:
It is common to want to prove convergence of a sequence
195:, respectively, and convergence is with respect to the
2591:
2427:
2367:
867:
to prove that it has a fixed point. This requires that
3307:
3268:
3194:
3165:
3145:
3125:
3093:
3064:
3018:
2998:
2881:
2846:
2826:
2768:
2748:
2729:{\displaystyle ||f(n)-f_{\infty }||_{f}\rightarrow 0}
2667:
2502:
2347:
2319:
2275:
2240:
2166:
2130:
2104:
2069:
2046:
2019:
1991:
1953:
1933:
1895:
1852:
1814:
1784:
1764:
1740:
1720:
1687:
1659:
1639:
1619:
1584:
1564:
1505:
1481:
1447:
1411:
1373:
1353:
1323:
1278:
1242:
1222:
1202:
1158:
1138:
1099:
1072:
1026:
1006:
986:
950:
873:
849:
806:
786:
728:
708:
649:
620:
600:
580:
521:
486:
459:
405:
372:
337:
317:
290:
270:
205:
171:
149:
106:
63:
1801:
480:and prove the required inequality. If the value of
19:are canonical patterns of mathematical proofs that
3313:
3293:
3209:
3180:
3151:
3131:
3111:
3079:
3053:
3031:
3004:
2984:
2861:
2832:
2813:{\displaystyle f_{n}(x)\rightarrow f_{\infty }(x)}
2812:
2754:
2728:
2653:
2536:
2484:
2413:
2353:
2331:
2305:
2261:
2226:
2151:
2116:
2090:
2052:
2025:
1997:
1977:
1939:
1919:
1881:
1829:
1790:
1770:
1746:
1726:
1702:
1665:
1645:
1625:
1605:
1570:
1550:
1487:
1459:
1433:
1397:
1359:
1329:
1309:
1264:
1228:
1208:
1188:
1144:
1112:
1085:
1058:
1012:
992:
972:
937:{\displaystyle \|T(x)-T(y)\|<\|\lambda (x-y)\|}
936:
855:
821:
792:
768:
714:
688:
635:
606:
586:
566:
499:
472:
445:
391:
358:
323:
303:
276:
238:
179:
157:
135:
92:
446:{\displaystyle \|f(k)-f_{\infty }\|<\epsilon }
3439:
2899:
2883:
2429:
2369:
3405:Functional Analysis: An Elementary Introduction
2654:{\textstyle \|g\|_{f}=\int _{x\in A}\|g(x)\|dx}
2566:If the inequality in the condition 2 is weak,
3087:. (Formally, a random variable is a mapping
1405:is a non-expansion mapping, but the sequence
2970:
2902:
2642:
2627:
2599:
2592:
2583:Convergence in the norm (strong convergence)
931:
910:
904:
874:
434:
406:
3421:
3390:Elementary Analysis: The Theory of Calculus
249:Useful approaches for this are as follows.
3358:
3294:{\displaystyle x_{n}:\Omega \rightarrow V}
1676:
1367:is a non-expansion mapping. For example,
3262:-- pointwise convergence of the mappings
3197:
3168:
3067:
2969:
2905:
2898:
1875:
1861:
1846:. In these cases, one defines a function
1817:
1690:
1653:is a non-expansion mapping, this implies
1341:
352:
173:
151:
123:
114:
80:
71:
1317:if the magnitudes of all eigenvalues of
1471:Contraction mappings on limited domains
510:
3440:
3349:Doob's martingale convergence theorems
3112:{\displaystyle x:\Omega \rightarrow V}
2661:is defined, and convergence occurs if
2227:{\displaystyle V(f(k+1))-V(f(k))<0}
829:by the square root of its magnitude.
34:There are many types of sequences and
2574:Convergence of sequences of functions
1710:has a convergent subsequence, by the
1066:, then it is sufficient to show that
359:{\displaystyle k_{0}\in \mathbb {N} }
31:when the argument tends to infinity.
3251:of the random variables to the limit
2361:is "radially unbounded", i.e., that
49:
2060:to be a Lyapunov function are that
252:
13:
3351:a random variable analogue of the
3308:
3282:
3126:
3100:
2938:
2893:
2796:
2742:-- convergence occurs if for each
2698:
2479:
2439:
2408:
2379:
2306:{\displaystyle {\dot {V}}(x)<0}
1551:{\displaystyle T(x)=\cos(\sin(x))}
1059:{\displaystyle T=T_{1}\circ T_{2}}
492:
429:
296:
14:
3459:
3402:
3301:to the limit, except at a set in
2005:satisfies the conditions to be a
1802:Monotonicity (Lyapunov functions)
776:, a matrix generalization of the
3387:
3247:-- pointwise convergence of the
3210:{\displaystyle \mathbb {R} ^{n}}
3181:{\displaystyle \mathbb {R} ^{n}}
3080:{\displaystyle \mathbb {R} ^{n}}
1830:{\displaystyle \mathbb {R} ^{n}}
1778:that contain no fixed points of
1703:{\displaystyle \mathbb {R} ^{n}}
3240:between functions is measured.
3054:Convergence of random variables
2548:in the study of convergence of
2035:ordinary differential equations
1120:are both contraction mappings.
973:{\displaystyle |\lambda |<1}
3415:
3396:
3381:
3285:
3225:, rather than the sequence of
3103:
2949:
2943:
2927:
2921:
2890:
2856:
2850:
2840:with the norm appropriate for
2807:
2801:
2788:
2785:
2779:
2720:
2710:
2704:
2687:
2681:
2674:
2669:
2639:
2633:
2568:LaSalle's invariance principle
2512:
2506:
2472:
2467:
2463:
2457:
2450:
2445:
2436:
2402:
2399:
2393:
2387:
2376:
2294:
2288:
2250:
2244:
2215:
2212:
2206:
2200:
2191:
2188:
2176:
2170:
2140:
2134:
2079:
2073:
1972:
1969:
1963:
1957:
1914:
1911:
1905:
1899:
1871:
1808:bounded monotonic sequence in
1600:
1585:
1545:
1542:
1536:
1527:
1515:
1509:
1428:
1422:
1383:
1377:
1292:
1279:
1259:
1253:
1168:
1162:
960:
952:
928:
916:
901:
895:
886:
880:
816:
810:
769:{\displaystyle f(k)=A^{k}f(0)}
763:
757:
738:
732:
683:
677:
665:
653:
630:
624:
567:{\displaystyle f(k+1)=T(f(k))}
561:
558:
552:
546:
537:
525:
418:
412:
239:{\displaystyle ||\cdot ||_{2}}
226:
220:
212:
207:
118:
75:
1:
3422:Billingsley, Patrick (1995).
3374:
3048:Convergence of Fourier series
3409:American Mathematics Society
3353:monotone convergence theorem
3321:with measure 0 in the limit.
2585:-- a function norm, such as
2557:delay differential equations
2537:{\displaystyle V(x)=x^{T}Ax}
689:{\displaystyle f(k+1)=Af(k)}
180:{\displaystyle \mathbb {R} }
158:{\displaystyle \mathbb {N} }
17:Convergence proof techniques
7:
3245:Convergence in distribution
1712:Bolzano–Weierstrass theorem
1633:for real arguments. Since
1613:, which is the codomain of
1310:{\displaystyle (I-A)^{-1}B}
500:{\displaystyle f_{\infty }}
304:{\displaystyle f_{\infty }}
10:
3464:
3255:Convergence in probability
3159:. The value space may be
2262:{\displaystyle f(k)\neq 0}
2040:The basic requirements on
1681:Every bounded sequence in
1673:is a contraction mapping.
1441:does not converge for any
1123:
834:Banach fixed-point theorem
392:{\displaystyle k>k_{0}}
2550:probability distributions
2091:{\displaystyle V(x)>0}
1189:{\displaystyle T(x)=Ax+B}
324:{\displaystyle \epsilon }
1434:{\displaystyle T^{n}(x)}
1265:{\displaystyle T^{k}(x)}
574:for some transformation
3424:Probability and Measure
3359:Topological convergence
3325:Convergence in the mean
3314:{\displaystyle \Omega }
3260:Almost sure convergence
3236:, depending on how the
3132:{\displaystyle \Omega }
2332:{\displaystyle x\neq 0}
2117:{\displaystyle x\neq 0}
1978:{\displaystyle V(f(k))}
1920:{\displaystyle V(f(k))}
1677:Convergent subsequences
1460:{\displaystyle x\neq 0}
1398:{\displaystyle T(x)=-x}
980:which is fixed for all
3315:
3295:
3249:distribution functions
3211:
3182:
3153:
3133:
3113:
3081:
3033:
3012:is the domain of each
3006:
2986:
2863:
2834:
2814:
2756:
2730:
2655:
2538:
2486:
2421:for any sequence with
2415:
2355:
2333:
2307:
2263:
2228:
2153:
2152:{\displaystyle V(0)=0}
2118:
2092:
2054:
2027:
1999:
1979:
1941:
1921:
1883:
1831:
1792:
1772:
1754:and that there are no
1748:
1728:
1704:
1667:
1647:
1627:
1607:
1572:
1552:
1489:
1461:
1435:
1399:
1361:
1342:Non-expansion mappings
1331:
1311:
1266:
1230:
1210:
1190:
1146:
1114:
1087:
1060:
1014:
994:
974:
938:
857:
823:
794:
770:
716:
690:
637:
608:
588:
568:
501:
474:
447:
393:
360:
325:
305:
278:
240:
181:
159:
137:
94:
3448:Mathematical analysis
3338:Dominated convergence
3316:
3296:
3212:
3183:
3154:
3134:
3114:
3082:
3034:
3032:{\displaystyle f_{n}}
3007:
2987:
2864:
2835:
2815:
2757:
2740:Pointwise convergence
2731:
2656:
2539:
2487:
2416:
2356:
2334:
2308:
2264:
2229:
2154:
2119:
2093:
2055:
2028:
2000:
1980:
1942:
1922:
1884:
1832:
1793:
1773:
1749:
1729:
1705:
1668:
1648:
1646:{\displaystyle \sin }
1628:
1626:{\displaystyle \sin }
1608:
1573:
1571:{\displaystyle \cos }
1553:
1490:
1462:
1436:
1400:
1362:
1332:
1312:
1267:
1231:
1211:
1191:
1147:
1115:
1113:{\displaystyle T_{2}}
1088:
1086:{\displaystyle T_{1}}
1061:
1015:
995:
975:
939:
858:
841:complete metric space
824:
795:
778:geometric progression
771:
717:
691:
638:
609:
589:
569:
502:
475:
473:{\displaystyle k_{0}}
448:
394:
361:
326:
306:
279:
241:
182:
160:
138:
95:
27:converge to a finite
3305:
3266:
3234:types of convergence
3192:
3163:
3143:
3123:
3119:from an event space
3091:
3062:
3016:
2996:
2879:
2862:{\displaystyle f(x)}
2844:
2824:
2766:
2746:
2665:
2589:
2500:
2425:
2365:
2345:
2317:
2273:
2238:
2164:
2128:
2102:
2067:
2044:
2017:
1989:
1951:
1931:
1893:
1850:
1837:converges to a limit
1812:
1782:
1762:
1738:
1718:
1685:
1657:
1637:
1617:
1582:
1562:
1503:
1479:
1445:
1409:
1371:
1351:
1321:
1276:
1240:
1220:
1200:
1156:
1136:
1097:
1070:
1024:
1004:
984:
948:
871:
847:
822:{\displaystyle f(k)}
804:
784:
726:
706:
647:
636:{\displaystyle f(k)}
618:
598:
578:
519:
511:Contraction mappings
484:
457:
403:
370:
335:
315:
288:
268:
203:
169:
147:
104:
61:
36:modes of convergence
3232:There are multiple
2873:uniform convergence
2561:Lyapunov-Krasovskii
2269:(discrete case) or
865:contraction mapping
3343:Carleson's theorem
3311:
3291:
3207:
3178:
3149:
3129:
3109:
3077:
3029:
3002:
2982:
2897:
2859:
2830:
2810:
2752:
2726:
2651:
2534:
2482:
2443:
2411:
2383:
2351:
2329:
2303:
2259:
2224:
2149:
2114:
2088:
2050:
2023:
2011:Lyapunov's theorem
1995:
1975:
1937:
1917:
1879:
1844:Aleksandr Lyapunov
1827:
1788:
1768:
1744:
1724:
1700:
1663:
1643:
1623:
1603:
1568:
1548:
1485:
1457:
1431:
1395:
1357:
1327:
1307:
1262:
1226:
1206:
1196:for some matrices
1186:
1142:
1110:
1083:
1056:
1010:
990:
970:
944:for some constant
934:
853:
819:
790:
780:. Alternatively,
766:
712:
686:
633:
604:
584:
564:
497:
470:
443:
389:
356:
321:
301:
274:
264:of convergence of
236:
177:
155:
133:
90:
3152:{\displaystyle V}
3139:to a value space
3005:{\displaystyle A}
2882:
2833:{\displaystyle x}
2755:{\displaystyle x}
2428:
2368:
2354:{\displaystyle V}
2339:(continuous case)
2285:
2053:{\displaystyle V}
2026:{\displaystyle f}
2007:Lyapunov function
1998:{\displaystyle V}
1940:{\displaystyle k}
1791:{\displaystyle T}
1771:{\displaystyle T}
1747:{\displaystyle T}
1727:{\displaystyle f}
1666:{\displaystyle T}
1488:{\displaystyle T}
1360:{\displaystyle T}
1330:{\displaystyle A}
1229:{\displaystyle B}
1209:{\displaystyle A}
1145:{\displaystyle T}
1013:{\displaystyle y}
993:{\displaystyle x}
856:{\displaystyle T}
836:(the domain is a
793:{\displaystyle T}
715:{\displaystyle A}
607:{\displaystyle T}
587:{\displaystyle T}
277:{\displaystyle f}
44:convergence tests
3455:
3432:
3431:
3419:
3413:
3412:
3400:
3394:
3393:
3385:
3320:
3318:
3317:
3312:
3300:
3298:
3297:
3292:
3278:
3277:
3216:
3214:
3213:
3208:
3206:
3205:
3200:
3187:
3185:
3184:
3179:
3177:
3176:
3171:
3158:
3156:
3155:
3150:
3138:
3136:
3135:
3130:
3118:
3116:
3115:
3110:
3086:
3084:
3083:
3078:
3076:
3075:
3070:
3038:
3036:
3035:
3030:
3028:
3027:
3011:
3009:
3008:
3003:
2991:
2989:
2988:
2983:
2956:
2952:
2942:
2941:
2920:
2919:
2896:
2868:
2866:
2865:
2860:
2839:
2837:
2836:
2831:
2819:
2817:
2816:
2811:
2800:
2799:
2778:
2777:
2761:
2759:
2758:
2753:
2735:
2733:
2732:
2727:
2719:
2718:
2713:
2707:
2702:
2701:
2677:
2672:
2660:
2658:
2657:
2652:
2626:
2625:
2607:
2606:
2543:
2541:
2540:
2535:
2527:
2526:
2491:
2489:
2488:
2483:
2475:
2470:
2453:
2448:
2442:
2420:
2418:
2417:
2412:
2382:
2360:
2358:
2357:
2352:
2338:
2336:
2335:
2330:
2312:
2310:
2309:
2304:
2287:
2286:
2278:
2268:
2266:
2265:
2260:
2233:
2231:
2230:
2225:
2158:
2156:
2155:
2150:
2123:
2121:
2120:
2115:
2097:
2095:
2094:
2089:
2059:
2057:
2056:
2051:
2032:
2030:
2029:
2024:
2004:
2002:
2001:
1996:
1984:
1982:
1981:
1976:
1946:
1944:
1943:
1938:
1927:is monotonic in
1926:
1924:
1923:
1918:
1888:
1886:
1885:
1880:
1878:
1870:
1869:
1864:
1836:
1834:
1833:
1828:
1826:
1825:
1820:
1797:
1795:
1794:
1789:
1777:
1775:
1774:
1769:
1753:
1751:
1750:
1745:
1733:
1731:
1730:
1725:
1709:
1707:
1706:
1701:
1699:
1698:
1693:
1672:
1670:
1669:
1664:
1652:
1650:
1649:
1644:
1632:
1630:
1629:
1624:
1612:
1610:
1609:
1606:{\displaystyle }
1604:
1577:
1575:
1574:
1569:
1558:. The function
1557:
1555:
1554:
1549:
1494:
1492:
1491:
1486:
1466:
1464:
1463:
1458:
1440:
1438:
1437:
1432:
1421:
1420:
1404:
1402:
1401:
1396:
1366:
1364:
1363:
1358:
1337:are less than 1.
1336:
1334:
1333:
1328:
1316:
1314:
1313:
1308:
1303:
1302:
1271:
1269:
1268:
1263:
1252:
1251:
1235:
1233:
1232:
1227:
1215:
1213:
1212:
1207:
1195:
1193:
1192:
1187:
1151:
1149:
1148:
1143:
1119:
1117:
1116:
1111:
1109:
1108:
1092:
1090:
1089:
1084:
1082:
1081:
1065:
1063:
1062:
1057:
1055:
1054:
1042:
1041:
1019:
1017:
1016:
1011:
999:
997:
996:
991:
979:
977:
976:
971:
963:
955:
943:
941:
940:
935:
862:
860:
859:
854:
828:
826:
825:
820:
799:
797:
796:
791:
775:
773:
772:
767:
753:
752:
721:
719:
718:
713:
695:
693:
692:
687:
642:
640:
639:
634:
613:
611:
610:
605:
594:. For example,
593:
591:
590:
585:
573:
571:
570:
565:
506:
504:
503:
498:
496:
495:
479:
477:
476:
471:
469:
468:
452:
450:
449:
444:
433:
432:
398:
396:
395:
390:
388:
387:
365:
363:
362:
357:
355:
347:
346:
330:
328:
327:
322:
311:is that for all
310:
308:
307:
302:
300:
299:
283:
281:
280:
275:
253:First principles
245:
243:
242:
237:
235:
234:
229:
223:
215:
210:
186:
184:
183:
178:
176:
164:
162:
161:
156:
154:
142:
140:
139:
134:
132:
131:
126:
117:
99:
97:
96:
91:
89:
88:
83:
74:
3463:
3462:
3458:
3457:
3456:
3454:
3453:
3452:
3438:
3437:
3436:
3435:
3420:
3416:
3403:Haase, Markus.
3401:
3397:
3388:Ross, Kenneth.
3386:
3382:
3377:
3361:
3306:
3303:
3302:
3273:
3269:
3267:
3264:
3263:
3201:
3196:
3195:
3193:
3190:
3189:
3172:
3167:
3166:
3164:
3161:
3160:
3144:
3141:
3140:
3124:
3121:
3120:
3092:
3089:
3088:
3071:
3066:
3065:
3063:
3060:
3059:
3056:
3023:
3019:
3017:
3014:
3013:
2997:
2994:
2993:
2937:
2933:
2915:
2911:
2910:
2906:
2886:
2880:
2877:
2876:
2845:
2842:
2841:
2825:
2822:
2821:
2795:
2791:
2773:
2769:
2767:
2764:
2763:
2747:
2744:
2743:
2714:
2709:
2708:
2703:
2697:
2693:
2673:
2668:
2666:
2663:
2662:
2615:
2611:
2602:
2598:
2590:
2587:
2586:
2576:
2522:
2518:
2501:
2498:
2497:
2471:
2466:
2449:
2444:
2432:
2426:
2423:
2422:
2372:
2366:
2363:
2362:
2346:
2343:
2342:
2318:
2315:
2314:
2277:
2276:
2274:
2271:
2270:
2239:
2236:
2235:
2165:
2162:
2161:
2129:
2126:
2125:
2103:
2100:
2099:
2068:
2065:
2064:
2045:
2042:
2041:
2018:
2015:
2014:
1990:
1987:
1986:
1952:
1949:
1948:
1932:
1929:
1928:
1894:
1891:
1890:
1874:
1865:
1860:
1859:
1851:
1848:
1847:
1821:
1816:
1815:
1813:
1810:
1809:
1804:
1783:
1780:
1779:
1763:
1760:
1759:
1739:
1736:
1735:
1719:
1716:
1715:
1694:
1689:
1688:
1686:
1683:
1682:
1679:
1658:
1655:
1654:
1638:
1635:
1634:
1618:
1615:
1614:
1583:
1580:
1579:
1563:
1560:
1559:
1504:
1501:
1500:
1480:
1477:
1476:
1473:
1446:
1443:
1442:
1416:
1412:
1410:
1407:
1406:
1372:
1369:
1368:
1352:
1349:
1348:
1344:
1322:
1319:
1318:
1295:
1291:
1277:
1274:
1273:
1247:
1243:
1241:
1238:
1237:
1221:
1218:
1217:
1201:
1198:
1197:
1157:
1154:
1153:
1137:
1134:
1133:
1126:
1104:
1100:
1098:
1095:
1094:
1077:
1073:
1071:
1068:
1067:
1050:
1046:
1037:
1033:
1025:
1022:
1021:
1005:
1002:
1001:
985:
982:
981:
959:
951:
949:
946:
945:
872:
869:
868:
848:
845:
844:
805:
802:
801:
785:
782:
781:
748:
744:
727:
724:
723:
707:
704:
703:
648:
645:
644:
619:
616:
615:
599:
596:
595:
579:
576:
575:
520:
517:
516:
513:
491:
487:
485:
482:
481:
464:
460:
458:
455:
454:
428:
424:
404:
401:
400:
383:
379:
371:
368:
367:
351:
342:
338:
336:
333:
332:
331:there exists a
316:
313:
312:
295:
291:
289:
286:
285:
269:
266:
265:
255:
230:
225:
224:
219:
211:
206:
204:
201:
200:
189:natural numbers
172:
170:
167:
166:
150:
148:
145:
144:
127:
122:
121:
113:
105:
102:
101:
84:
79:
78:
70:
62:
59:
58:
55:
50:Convergence in
12:
11:
5:
3461:
3451:
3450:
3434:
3433:
3414:
3395:
3379:
3378:
3376:
3373:
3360:
3357:
3356:
3355:
3346:
3340:
3328:
3327:
3322:
3310:
3290:
3287:
3284:
3281:
3276:
3272:
3257:
3252:
3204:
3199:
3175:
3170:
3148:
3128:
3108:
3105:
3102:
3099:
3096:
3074:
3069:
3055:
3052:
3051:
3050:
3041:
3040:
3026:
3022:
3001:
2981:
2978:
2975:
2972:
2968:
2965:
2962:
2959:
2955:
2951:
2948:
2945:
2940:
2936:
2932:
2929:
2926:
2923:
2918:
2914:
2909:
2904:
2901:
2895:
2892:
2889:
2885:
2870:
2858:
2855:
2852:
2849:
2829:
2809:
2806:
2803:
2798:
2794:
2790:
2787:
2784:
2781:
2776:
2772:
2751:
2737:
2725:
2722:
2717:
2712:
2706:
2700:
2696:
2692:
2689:
2686:
2683:
2680:
2676:
2671:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2624:
2621:
2618:
2614:
2610:
2605:
2601:
2597:
2594:
2575:
2572:
2533:
2530:
2525:
2521:
2517:
2514:
2511:
2508:
2505:
2494:
2493:
2481:
2478:
2474:
2469:
2465:
2462:
2459:
2456:
2452:
2447:
2441:
2438:
2435:
2431:
2410:
2407:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2381:
2378:
2375:
2371:
2350:
2340:
2328:
2325:
2322:
2302:
2299:
2296:
2293:
2290:
2284:
2281:
2258:
2255:
2252:
2249:
2246:
2243:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2159:
2148:
2145:
2142:
2139:
2136:
2133:
2113:
2110:
2107:
2087:
2084:
2081:
2078:
2075:
2072:
2049:
2022:
1994:
1985:converges. If
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1936:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1877:
1873:
1868:
1863:
1858:
1855:
1824:
1819:
1803:
1800:
1787:
1767:
1756:invariant sets
1743:
1723:
1697:
1692:
1678:
1675:
1662:
1642:
1622:
1602:
1599:
1596:
1593:
1590:
1587:
1567:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1508:
1484:
1472:
1469:
1456:
1453:
1450:
1430:
1427:
1424:
1419:
1415:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1356:
1343:
1340:
1339:
1338:
1326:
1306:
1301:
1298:
1294:
1290:
1287:
1284:
1281:
1261:
1258:
1255:
1250:
1246:
1225:
1205:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1141:
1125:
1122:
1107:
1103:
1080:
1076:
1053:
1049:
1045:
1040:
1036:
1032:
1029:
1009:
989:
969:
966:
962:
958:
954:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
852:
818:
815:
812:
809:
789:
765:
762:
759:
756:
751:
747:
743:
740:
737:
734:
731:
711:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
655:
652:
632:
629:
626:
623:
603:
583:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
512:
509:
494:
490:
467:
463:
442:
439:
436:
431:
427:
423:
420:
417:
414:
411:
408:
386:
382:
378:
375:
354:
350:
345:
341:
320:
298:
294:
273:
254:
251:
233:
228:
222:
218:
214:
209:
197:Euclidean norm
175:
153:
130:
125:
120:
116:
112:
109:
87:
82:
77:
73:
69:
66:
54:
48:
9:
6:
4:
3:
2:
3460:
3449:
3446:
3445:
3443:
3429:
3425:
3418:
3410:
3406:
3399:
3391:
3384:
3380:
3372:
3370:
3366:
3354:
3350:
3347:
3344:
3341:
3339:
3336:
3335:
3334:
3331:
3326:
3323:
3288:
3279:
3274:
3270:
3261:
3258:
3256:
3253:
3250:
3246:
3243:
3242:
3241:
3239:
3235:
3230:
3228:
3224:
3223:distributions
3220:
3202:
3173:
3146:
3106:
3097:
3094:
3072:
3049:
3046:
3045:
3044:
3024:
3020:
2999:
2979:
2976:
2973:
2966:
2963:
2960:
2957:
2953:
2946:
2934:
2930:
2924:
2916:
2912:
2907:
2887:
2874:
2871:
2853:
2847:
2827:
2804:
2792:
2782:
2774:
2770:
2749:
2741:
2738:
2723:
2715:
2694:
2690:
2684:
2678:
2648:
2645:
2636:
2630:
2622:
2619:
2616:
2612:
2608:
2603:
2595:
2584:
2581:
2580:
2579:
2571:
2570:may be used.
2569:
2564:
2563:functionals.
2562:
2558:
2553:
2551:
2547:
2531:
2528:
2523:
2519:
2515:
2509:
2503:
2476:
2460:
2454:
2433:
2405:
2396:
2390:
2384:
2373:
2348:
2341:
2326:
2323:
2320:
2300:
2297:
2291:
2282:
2279:
2256:
2253:
2247:
2241:
2221:
2218:
2209:
2203:
2197:
2194:
2185:
2182:
2179:
2173:
2167:
2160:
2146:
2143:
2137:
2131:
2111:
2108:
2105:
2085:
2082:
2076:
2070:
2063:
2062:
2061:
2047:
2038:
2036:
2020:
2013:implies that
2012:
2008:
1992:
1966:
1960:
1954:
1934:
1908:
1902:
1896:
1866:
1856:
1853:
1845:
1840:
1838:
1822:
1799:
1785:
1765:
1757:
1741:
1721:
1713:
1695:
1674:
1660:
1640:
1620:
1597:
1594:
1591:
1588:
1565:
1539:
1533:
1530:
1524:
1521:
1518:
1512:
1506:
1498:
1482:
1468:
1454:
1451:
1448:
1425:
1417:
1413:
1392:
1389:
1386:
1380:
1374:
1354:
1324:
1304:
1299:
1296:
1288:
1285:
1282:
1272:converges to
1256:
1248:
1244:
1223:
1203:
1183:
1180:
1177:
1174:
1171:
1165:
1159:
1152:has the form
1139:
1131:
1130:
1129:
1121:
1105:
1101:
1078:
1074:
1051:
1047:
1043:
1038:
1034:
1030:
1027:
1007:
987:
967:
964:
956:
925:
922:
919:
913:
907:
898:
892:
889:
883:
877:
866:
850:
842:
839:
835:
830:
813:
807:
787:
779:
760:
754:
749:
745:
741:
735:
729:
709:
702:
699:
680:
674:
671:
668:
662:
659:
656:
650:
627:
621:
601:
581:
555:
549:
543:
540:
534:
531:
528:
522:
508:
488:
465:
461:
440:
437:
425:
421:
415:
409:
384:
380:
376:
373:
366:such for all
348:
343:
339:
318:
292:
271:
263:
260:
250:
247:
231:
216:
198:
194:
190:
187:refer to the
128:
110:
107:
85:
67:
64:
53:
47:
45:
41:
37:
32:
30:
26:
22:
18:
3423:
3417:
3404:
3398:
3389:
3383:
3362:
3332:
3329:
3237:
3231:
3226:
3222:
3218:
3057:
3042:
2577:
2565:
2554:
2495:
2039:
1841:
1805:
1680:
1474:
1345:
1127:
831:
514:
256:
248:
193:real numbers
100:or function
56:
51:
33:
16:
15:
3428:John Wesley
3392:. Springer.
698:conformable
284:to a limit
3375:References
1889:such that
722:, so that
614:could map
262:definition
3369:Hausdorff
3333:See also
3309:Ω
3286:→
3283:Ω
3221:, or the
3219:functions
3127:Ω
3104:→
3101:Ω
3043:See also
2964:∈
2939:∞
2931:−
2894:∞
2891:→
2797:∞
2789:→
2721:→
2699:∞
2691:−
2643:‖
2628:‖
2620:∈
2613:∫
2600:‖
2593:‖
2546:entropies
2480:∞
2440:∞
2437:→
2409:∞
2380:∞
2377:→
2324:≠
2283:˙
2254:≠
2195:−
2109:≠
1947:and thus
1872:→
1589:−
1534:
1525:
1452:≠
1390:−
1297:−
1286:−
1044:∘
957:λ
932:‖
923:−
914:λ
911:‖
905:‖
890:−
875:‖
838:non-empty
696:for some
493:∞
441:ϵ
435:‖
430:∞
422:−
407:‖
349:∈
319:ϵ
297:∞
217:⋅
119:→
76:→
25:functions
21:sequences
3442:Category
3365:topology
3238:distance
2098:for all
1497:codomain
259:analytic
191:and the
143:, where
1236:, then
1124:Example
3227:values
2992:where
1806:Every
701:matrix
40:series
2009:then
863:is a
29:limit
2555:For
2313:for
2298:<
2234:for
2219:<
2124:and
2083:>
1216:and
1093:and
1000:and
965:<
908:<
438:<
377:>
257:The
165:and
3229:.)
2900:sup
2884:lim
2430:lim
2370:lim
1758:of
1641:sin
1621:sin
1566:cos
1531:sin
1522:cos
1475:If
1132:If
643:to
23:or
3444::
3426:.
3407:.
2762:,
2552:.
1839:.
399:,
246:.
199:,
46:.
3430:.
3411:.
3289:V
3280::
3275:n
3271:x
3203:n
3198:R
3174:n
3169:R
3147:V
3107:V
3098::
3095:x
3073:n
3068:R
3039:.
3025:n
3021:f
3000:A
2980:,
2977:0
2974:=
2971:}
2967:A
2961:x
2958::
2954:|
2950:)
2947:x
2944:(
2935:f
2928:)
2925:x
2922:(
2917:n
2913:f
2908:|
2903:{
2888:n
2869:.
2857:)
2854:x
2851:(
2848:f
2828:x
2808:)
2805:x
2802:(
2793:f
2786:)
2783:x
2780:(
2775:n
2771:f
2750:x
2724:0
2716:f
2711:|
2705:|
2695:f
2688:)
2685:n
2682:(
2679:f
2675:|
2670:|
2649:x
2646:d
2640:)
2637:x
2634:(
2631:g
2623:A
2617:x
2609:=
2604:f
2596:g
2532:x
2529:A
2524:T
2520:x
2516:=
2513:)
2510:x
2507:(
2504:V
2492:.
2477:=
2473:|
2468:|
2464:)
2461:k
2458:(
2455:f
2451:|
2446:|
2434:k
2406:=
2403:)
2400:)
2397:k
2394:(
2391:f
2388:(
2385:V
2374:k
2349:V
2327:0
2321:x
2301:0
2295:)
2292:x
2289:(
2280:V
2257:0
2251:)
2248:k
2245:(
2242:f
2222:0
2216:)
2213:)
2210:k
2207:(
2204:f
2201:(
2198:V
2192:)
2189:)
2186:1
2183:+
2180:k
2177:(
2174:f
2171:(
2168:V
2147:0
2144:=
2141:)
2138:0
2135:(
2132:V
2112:0
2106:x
2086:0
2080:)
2077:x
2074:(
2071:V
2048:V
2021:f
1993:V
1973:)
1970:)
1967:k
1964:(
1961:f
1958:(
1955:V
1935:k
1915:)
1912:)
1909:k
1906:(
1903:f
1900:(
1897:V
1876:R
1867:n
1862:R
1857::
1854:V
1823:n
1818:R
1786:T
1766:T
1742:T
1722:f
1696:n
1691:R
1661:T
1601:]
1598:1
1595:,
1592:1
1586:[
1546:)
1543:)
1540:x
1537:(
1528:(
1519:=
1516:)
1513:x
1510:(
1507:T
1483:T
1455:0
1449:x
1429:)
1426:x
1423:(
1418:n
1414:T
1393:x
1387:=
1384:)
1381:x
1378:(
1375:T
1355:T
1325:A
1305:B
1300:1
1293:)
1289:A
1283:I
1280:(
1260:)
1257:x
1254:(
1249:k
1245:T
1224:B
1204:A
1184:B
1181:+
1178:x
1175:A
1172:=
1169:)
1166:x
1163:(
1160:T
1140:T
1106:2
1102:T
1079:1
1075:T
1052:2
1048:T
1039:1
1035:T
1031:=
1028:T
1008:y
988:x
968:1
961:|
953:|
929:)
926:y
920:x
917:(
902:)
899:y
896:(
893:T
887:)
884:x
881:(
878:T
851:T
817:)
814:k
811:(
808:f
788:T
764:)
761:0
758:(
755:f
750:k
746:A
742:=
739:)
736:k
733:(
730:f
710:A
684:)
681:k
678:(
675:f
672:A
669:=
666:)
663:1
660:+
657:k
654:(
651:f
631:)
628:k
625:(
622:f
602:T
582:T
562:)
559:)
556:k
553:(
550:f
547:(
544:T
541:=
538:)
535:1
532:+
529:k
526:(
523:f
489:f
466:0
462:k
426:f
419:)
416:k
413:(
410:f
385:0
381:k
374:k
353:N
344:0
340:k
293:f
272:f
232:2
227:|
221:|
213:|
208:|
174:R
152:N
129:n
124:R
115:R
111::
108:f
86:n
81:R
72:N
68::
65:f
52:R
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