Levi-Civita field
E627727
The Levi-Civita field is a non-Archimedean ordered field of formal power series with real coefficients and well-ordered rational exponents, used to rigorously model infinitesimals and infinite quantities in analysis.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
Hahn field
ⓘ
non-Archimedean field ⓘ ordered field ⓘ real-closed field ⓘ valued field ⓘ |
| contains | infinitesimal neighborhoods of real numbers ⓘ |
| extends | field of real numbers ⓘ |
| hasAnalyticUse | definition of derivatives via infinitesimal quotients ⓘ |
| hasAnalyticUse | definition of integrals via infinitesimal partitions ⓘ |
| hasArchimedeanProperty | fails the Archimedean property ⓘ |
| hasCharacteristic | 0 ⓘ |
| hasCoefficientField | real numbers ⓘ |
| hasComparison |
is different from Robinson’s hyperreal field
ⓘ
provides an alternative to nonstandard analysis ⓘ |
| hasConstruction | formal power series with real coefficients and well-ordered rational exponents ⓘ |
| hasElementForm | sum over q in Q of a_q t^q with well-ordered support ⓘ |
| hasElements | formal power series ⓘ |
| hasExponentCondition | supports are well-ordered subsets of the rationals ⓘ |
| hasExponentGroup | rational numbers ⓘ |
| hasInfiniteElementProperty | every infinitely large element is larger than any real number ⓘ |
| hasInfinitesimal | t with 0 < t < 1/n for all positive integers n ⓘ |
| hasInfinitesimalProperty | every positive infinitesimal is smaller than any positive real number ⓘ |
| hasOrdering | lexicographic order on exponents and coefficients ⓘ |
| hasProperty |
Cauchy complete with respect to its natural valuation
ⓘ
contains infinitely large elements ⓘ contains infinitesimal elements ⓘ isomorphic copies of the real numbers embed as constant series ⓘ non-Archimedean ⓘ real-closed ⓘ spherically complete ⓘ strictly extends the real numbers ⓘ totally ordered ⓘ |
| hasSeriesSupportProperty | each nonzero element has a least exponent in its support ⓘ |
| hasSubstructure | copy of the rational numbers as constant series with rational coefficients ⓘ |
| hasTopology | valuation topology induced by its natural valuation ⓘ |
| hasUse |
asymptotic analysis
ⓘ
generalized differential calculus ⓘ non-Archimedean functional analysis ⓘ nonstandard-style analysis without ultrafilters ⓘ rigorous modeling of infinite quantities ⓘ rigorous modeling of infinitesimals ⓘ |
| hasValuation | map sending nonzero series to least exponent with nonzero coefficient ⓘ |
| isSubfieldOf | field of Hahn series with real coefficients and rational exponents ⓘ |
| namedAfter | Tullio Levi-Civita NERFINISHED ⓘ |
| supports |
Cauchy product of series for multiplication
ⓘ
termwise addition of series ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.