33:
1070:
Every normal convergent series is uniformly convergent, locally uniformly convergent, and compactly uniformly convergent. This is very important, since it assures that any re-arrangement of the series, any derivatives or integrals of the series, and sums and products with other convergent series will
429:
621:
724:, i.e. convergence of the partial sum sequence in the topology induced by the uniform norm. Normal convergence implies norm-topology convergence if and only if the space of functions under consideration is
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1217:
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with respect to the uniform norm. (The converse does not hold even for complete function spaces: for example, consider the harmonic series as a sequence of constant functions).
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and thus diverges. An example using continuous functions can be made by replacing these functions with bump functions of height 1/
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424:{\displaystyle \sum _{n=0}^{\infty }\|f_{n}\|:=\sum _{n=0}^{\infty }\sup _{x\in S}|f_{n}(x)|<\infty .}
1336:
1061:(even in the weakest sense), local normal convergence and compact normal convergence are equivalent.
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152:, it has the useful property that it is preserved when the order of summation is changed.
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161:
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293:
17:
616:{\displaystyle f_{n}(x)={\begin{cases}1/n,&x=n,\\0,&x\neq n.\end{cases}}}
130:
32:
514:. However, they should not be confused; to illustrate this, consider
443:, i.e., uniform convergence of the series of nonnegative functions
951:
A series is said to be "normally convergent on compact subsets of
26:
896:{\displaystyle \sum _{n=0}^{\infty }\|f_{n}\|_{U}<\infty }
609:
720:
As well, normal convergence of a series is different from
160:
The concept of normal convergence was first introduced by
741:
A series can be called "locally normally convergent on
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1278:{\displaystyle \sum _{n=0}^{\infty }f_{\tau (n)}(x)}
1033:{\displaystyle \sum _{n=0}^{\infty }f_{n}\mid _{K}}
823:{\displaystyle \sum _{n=0}^{\infty }f_{n}\mid _{U}}
57:. Unsourced material may be challenged and removed.
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1212:{\displaystyle \tau :\mathbb {N} \to \mathbb {N} }
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713:and width 1 centered at each natural number
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686:{\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|}
503:{\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|}
166:Leçons sur les théories générales de l'analyse
1124:{\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}
278:{\displaystyle \sum _{n=0}^{\infty }f_{n}(x)}
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920:
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296:of the terms of the series converges, i.e.,
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117:Learn how and when to remove this message
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955:" or "compactly normally convergent on
833:is normally convergent, i.e. such that
214:{\displaystyle f_{n}:S\to \mathbb {C} }
1316:Modes of convergence (annotated index)
939:is the supremum over the domain
7:
1176:also converges normally to the same
55:adding citations to reliable sources
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1005:
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757:such that the series of functions
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693:is uniformly convergent (for any
1071:converge to the "correct" value.
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1043:is normally convergent on
510:; this fact is essentially the
42:needs additional citations for
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959:" if for every compact subset
932:{\displaystyle \|\cdot \|_{U}}
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441:uniform absolute convergence
1342:Encyclopedia of Mathematics
1335:Solomentsev, E.D. (2001) ,
439:Normal convergence implies
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1285:is normally convergent to
1131:is normally convergent to
967:, the series of functions
947:Compact normal convergence
1381:Convergence (mathematics)
766:restricted to the domain
722:norm-topology convergence
737:Local normal convergence
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1376:Mathematical analysis
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1180:. That is, for every
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1337:"Normal convergence"
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164:in 1908 in his book
150:absolute-convergence
66:"Normal convergence"
51:improve this article
753:has a neighborhood
290:normally convergent
223:normed vector space
18:Normally convergent
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512:Weierstrass M-test
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134:normal convergence
1298:{\displaystyle f}
1144:{\displaystyle f}
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292:if the series of
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16:(Redirected from
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732:Generalizations
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294:uniform norms
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136:is a type of
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107:December 2012
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96:
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89:
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78:
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71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
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438:
435:Distinctions
289:
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177:
176:Given a set
175:
165:
159:
133:
128:
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
221:(or to any
138:convergence
131:mathematics
1370:Categories
1356:1402006098
1322:References
1065:Properties
288:is called
172:Definition
162:René Baire
77:newspapers
1347:EMS Press
1253:τ
1243:∞
1228:∑
1202:→
1191:τ
1182:bijection
1098:∞
1083:∑
1022:∣
1006:∞
991:∑
921:‖
917:⋅
914:‖
891:∞
879:‖
865:‖
860:∞
845:∑
812:∣
796:∞
781:∑
650:∞
635:∑
598:≠
467:∞
452:∑
416:∞
376:∈
363:∞
348:∑
341:‖
328:‖
323:∞
308:∑
252:∞
237:∑
204:→
146:functions
1310:See also
726:complete
148:. Like
156:History
91:scholar
1353:
759:ƒ
703:ε
695:ε
142:series
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697:take
98:JSTOR
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1351:ISBN
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888:<
701:≥ 1/
413:<
140:for
70:news
1074:If
1057:is
963:of
749:in
369:sup
144:of
129:In
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1178:ƒ
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