1968:
718:
by its smallest closed subspace containing all the values of the crinkled arc; b) uniform scalings; c) translations; d) reparametrizations. Now use these normalizations to define an equivalence relation on crinkled arcs if any two of them become identical after any sequence of such normalizations.
483:
turn during the passage between the chords' farthest end-points" and observes that such a curve would "seem to be making a sudden right angle turn at each point" which would justify the choice of terminology. Halmos deduces that such a curve could not have a
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Writing in 1975, Richard Vitale considers Halmos's empirical observation that every attempt to construct a crinkled arc results in essentially the same solution and
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at any point, and uses the concept to justify his statement that an infinite-dimensional
Hilbert space is "even roomier than it looks".
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661:{\displaystyle f(t)={\sqrt {2}}\,\sum _{n=1}^{\infty }x_{n}{\frac {\sin(n-1/2)\pi t}{(n-1/2)\pi }}}
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Then there is just one equivalence class, and Vitale's formula describes a canonical example.
714:. The normalizations that need to be allowed are the following: a) Replace the Hilbert space
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787:, Graduate Texts in Mathematics, vol. 19, Springer-Verlag,
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Vitale, Richard A. (1975), "Representation of a crinkled arc",
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Differentiable vectorâvalued functions from
Euclidean space
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is a crinkled arc if it is continuous and possesses the
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Pages displaying wikidata descriptions as a fallback
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35:. The concept is usually credited to
1274:infinite-dimensional Gaussian measure
7:
1145:Infinite-dimensional vector function
528:, after appropriate normalizations,
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14:
1212:Generalizations of the derivative
1178:Differentiation in Fréchet spaces
979:Compact operator on Hilbert space
826:10.1090/S0002-9939-1975-0388056-1
20:, and in particular the study of
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394:{\displaystyle f(d)-f(c)}
350:{\displaystyle f(b)-f(a)}
1801:Spectrum of a C*-algebra
1371:Inverse function theorem
1258:Classical Wiener measure
783:Halmos, Paul R. (1982),
109:is a Hilbert space with
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42:Specifically, consider
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748:(8 November 1982).
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946:
945:
943:
941:Other results
939:
933:
930:
928:
925:
923:
920:
919:
917:
913:
907:
904:
902:
899:
897:
893:
892:Hilbert space
890:
888:
884:
883:Inner product
881:
879:
876:
875:
873:
869:
865:
858:
853:
851:
846:
844:
839:
838:
835:
827:
822:
818:
814:
813:
807:
804:
798:
794:
790:
786:
781:
777:
773:
769:
763:
759:
755:
751:
747:
743:
742:
738:
730:
727:
726:
722:
720:
717:
713:
695:
690:
685:
681:
677:
652:
646:
642:
638:
635:
632:
624:
621:
615:
611:
607:
604:
601:
595:
592:
584:
580:
569:
566:
563:
559:
552:
547:
541:
535:
527:
508:
502:
494:
489:
487:
482:
477:
475:
456:
453:
450:
424:
421:
418:
408:
405:whenever the
404:
385:
379:
376:
370:
364:
341:
335:
332:
326:
320:
313:
310:that is, the
297:
294:
291:
282:
276:
273:
267:
261:
258:
252:
246:
243:
237:
231:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
171:property: if
170:
154:
134:
128:
125:
122:
112:
111:inner product
96:
76:
73:
64:
61:
58:
52:
49:
40:
38:
34:
31:
28:is a type of
27:
23:
19:
1934:Balanced set
1908:Distribution
1846:Applications
1699:KreinâMilman
1684:Closed graph
1456:Applications
1414:Crinkled arc
1413:
1350:PaleyâWiener
1056:
1047:
1043:
1039:
1035:
1004:Self-adjoint
915:Main results
816:
810:
784:
749:
715:
490:
478:
168:
147:We say that
41:
26:crinkled arc
25:
15:
1863:Heat kernel
1853:Hardy space
1760:Trace class
1674:HahnâBanach
1636:Topological
1222:Holomorphic
1205:Directional
1165:Derivatives
1014:Trace class
819:: 303â304,
481:right-angle
37:Paul Halmos
18:mathematics
1987:Categories
1796:C*-algebra
1611:Properties
739:References
403:orthogonal
30:continuous
1770:Unbounded
1765:Transpose
1723:Operators
1652:Separable
1647:Reflexive
1632:Algebraic
1618:Barrelled
1345:Regulated
1317:Integrals
653:π
636:−
622:π
605:−
596:
575:∞
560:∑
407:intervals
377:−
333:−
289:⟩
274:−
244:−
229:⟨
206:≤
194:≤
182:≤
132:⟩
129:⋅
123:⋅
120:⟨
71:→
53::
1972:Category
1784:Algebras
1666:Theorems
1623:Complete
1592:Schwartz
1538:glossary
1299:Strongly
1100:Analysis
1028:Examples
723:See also
1775:Unitary
1755:Nuclear
1740:Compact
1735:Bounded
1730:Adjoint
1704:Minâmax
1597:Sobolev
1582:Nuclear
1572:Hilbert
1567:Fréchet
1532: (
1465: (
1407:Related
1359:Results
1335:Dunford
1325:Bochner
1291:Bochner
1265:Measure
1042:) with
1019:Unitary
878:Adjoint
776:8169781
486:tangent
169:crinkly
1750:Normal
1587:Orlicz
1577:Hölder
1557:Banach
1546:Spaces
1534:topics
1467:bundle
1295:Weakly
1284:Vector
999:Normal
799:
774:
764:
710:is an
668:where
493:proves
312:chords
89:where
1562:Besov
1188:Total
1050:<â
495:that
221:then
33:curve
1910:(or
1628:Dual
972:Maps
894:and
885:and
797:ISBN
772:OCLC
762:ISBN
472:are
440:and
401:are
357:and
200:<
188:<
24:, a
1102:in
821:doi
789:doi
593:sin
16:In
1989::
1536:â
1297:/
1293:/
817:52
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795:,
770:.
760:.
752:.
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1630:(
1540:)
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1057:F
1048:n
1044:K
1040:K
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960:(
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