Épigraphe (mathématiques)

Pour les articles homonymes, voir Épigraphe.

Cet article est une ébauche concernant l’analyse.

Vous pouvez partager vos connaissances en l’améliorant (comment ?) selon les recommandations des projets correspondants.

Soit $f$ une fonction définie sur un ensemble $\mathbb {E}$ à valeurs dans la droite réelle achevée ${\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,+\infty \}$ . L'épigraphe de $f$ est l'ensemble noté $\operatorname {epi} \,f$ et défini par

\operatorname {epi} \,f:=\{(x,\alpha )\in \mathbb {E} \times \mathbb {R} :f(x)\leqslant \alpha \}.

Il s'agit donc de l'ensemble des points de l'ensemble produit $\mathbb {E} \times \mathbb {R}$ qui sont situés au-dessus du graphe de $f$ (épi venant du grec ancien et signifiant sur, au-dessus).

L'épigraphe strict de $f$ est l'ensemble noté $\operatorname {epi} _{s}\,f$ et défini par

$\operatorname {epi} _{s}\,f:=\{(x,\alpha )\in \mathbb {E} \times \mathbb {R} :f(x)<\alpha \}.$

Exemples d'utilisation

L'épigraphe permet de transférer aux fonctions des notions définies pour les ensembles. En voici deux exemples.

Si $\mathbb {E}$ est un espace topologique, on démontre qu'une application $f:\mathbb {E} \to {\overline {\mathbb {R} }}$ est semi-continue inférieurement si et seulement si son épigraphe est un fermé de l'espace topologique produit $\mathbb {E} \times \mathbb {R}$ . En analyse convexe, une telle application est dite fermée^[1].
Si $\mathbb {E}$ est un espace vectoriel réel, on dit^[2] que $f:\mathbb {E} \to {\overline {\mathbb {R} }}$ est convexe si son épigraphe est un convexe de l'espace vectoriel produit $\mathbb {E} \times \mathbb {R}$ .

Notes et références

↑ Cette notion ne doit pas être confondue avec celle d'application fermée en topologie générale.
↑ (en) Charalambos D. Aliprantis et Kim C. Border, Infinite Dimensional Analysis : A Hitchhiker's Guide, Springer, 2007, 3^e éd. (lire en ligne), p. 254.

Read other articles:

Tsu Shi Ma Mi Re

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Desember 2023. Tsu Shi Ma Mi ReつしまみれInformasi latar belakangAsalPrefektur Chiba, JapanGenreRockTahun aktif1999–sekarangLabelBenten Label Tokyo, July, Australian Cattle GodSitus webwww.tsushimamire.comAnggotaMari, Mizue, Yayoi Tsu Shi Ma Mi Re (つしま�...

Good Night, Paul

Good Night, PaulPoster filmSutradaraWalter EdwardsProduserLewis J. SelznickDitulis olehCharles Dickson (cerita)Roland Oliver (cerita)Julia Crawford Ivers (adaptasi)PemeranHarrison FordConstance TalmadgeNorman KerryPenata musikHarry B. OlsenSinematograferJames Van TreesDistributorSelect PicturesTanggal rilis 20 Juni 1918 (1918-06-20) DurasiLima rolNegaraAmerika SerikatBahasaBisu (intertitel Inggris) Harrison Ford, Constance Talmadge, dan John Steppling Good Night, Paul adalah sebuah...

Danau Beyşehir

Danau BeyşehirCitra satelit Danau BeyşehirKoordinat37°47′0″N 31°33′0″E / 37.78333°N 31.55000°E / 37.78333; 31.55000Koordinat: 37°47′0″N 31°33′0″E / 37.78333°N 31.55000°E / 37.78333; 31.55000Terletak di negaraTurkiPanjang maksimal45 km (28 mil)Lebar maksimal20 km (12 mil)Area permukaan650,00 km2 (250,97 mil persegi)Ketinggian permukaan1.123 m (3.684 kaki)Danau Beyşehir (Turkish: Beyşehir G...

رونه بيدرسون

رونه بيدرسون معلومات شخصية الميلاد 19 مايو 1963 (العمر 60 سنة)موس الجنسية النرويج الجوائز جائزة المتلاعب الشرفية [لغات أخرى]‏ (1998) تعديل مصدري - تعديل روني بيدرسون، حكم كرة قدم نرويجي دولي.[1][2][3] و قد حكم في كأس الخليج 2002، وقد أدار مباراة منتخب ا...

Mohamed Eisa

Sudanese professional footballer Mohamed Eisa Eisa playing for Greenwich Borough in October 2016.Personal informationFull name Mohamed Mamoun Eisa[1]Date of birth (1994-07-12) 12 July 1994 (age 29)[2]Place of birth Khartoum, SudanHeight 1.83 m (6 ft 0 in)[3]Position(s) ForwardTeam informationCurrent team Exeter City(on loan from Milton Keynes Dons)Number 27Youth career Pro Touch Soccer AcademySenior career*Years Team Apps (Gls)2012–2015 Dartford 2...

Author

Creator of an original work For other uses, see Author (disambiguation). This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these template messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Author – news · newspapers · books · sch...

Lunar Module Eagle

Lunar lander used for Apollo 11 EaglePart of Apollo 11Eagle at Tranquility Base, July 20, 1969.Neil Armstrong photographs Buzz Aldrin.TypeLunar moduleClassApollo Lunar ModuleManufacturerGrummanSpecificationsLaunch mass33,294.5 lb (15,102.1 kg)[1]HistoryFateAscent stage: Abandoned in lunar orbit; impact location unknownDescent stage: Landed at Tranquility Base; still there Part of a series onApollo 11 Crew Neil Armstrong · Buzz Aldrin · Michael Collins Spacecraft CM-107 Col...

Manu Sánchez (calciatore 2000)

Manu Sánchez Nazionalità Spagna Altezza 179 cm Peso 70 kg Calcio Ruolo Difensore Squadra Celta Vigo Carriera Giovanili 201?-2014 Cornellà2014-2019 Atlético Madrid Squadre di club1 2019-2020 Atlético Madrid B29 (0)[1]2019-2021 Atlético Madrid6 (0)2021-2023→ Osasuna80 (1)2023- Celta Vigo17 (0) Nazionale 2020- Spagna U-216 (2) Palmarès Europei di calcio Under-21 Argento Romania-Georgia 2023 1 I due numeri indicano le presenze e l...

Katrineholm

Place in Södermanland, SwedenKatrineholmKatrineholm Stora torgetKatrineholmShow map of SödermanlandKatrineholmShow map of SwedenCoordinates: 59°00′N 16°12′E / 59.000°N 16.200°E / 59.000; 16.200CountrySwedenProvinceSödermanlandCountySödermanland CountyMunicipalityKatrineholm MunicipalityArea[1] • Total11.46 km2 (4.42 sq mi)Population (2018)[1] • Total24,271 • Density1,919/km2 (4,970/sq&#...

Ketika Cinta Bertasbih 2

Ketika Cinta Bertasbih 2SutradaraChaerul UmamProduser Mitzy Christina Cindy Christina SkenarioImam TantowiBerdasarkanKetika Cinta Bertasbiholeh Habiburrahman El ShirazyPemeran Kholidi Asadil Alam Oki Setiana Dewi Alice Sofie Norin Andi Arsyil Rahman Meyda Sefira Penata musik Anto Hoed Melly Goeslaw SinematograferRudy KurwetPenyuntingRizal BasriPerusahaanproduksiSinemArt PicturesTanggal rilis 17 September 2009 (2009-09-17) (Indonesia) Durasi120 menitNegaraIndonesiaBahasaBahasa I...

Roanoke College

Private liberal arts college in Salem, Virginia, USA Roanoke CollegeFormer namesVirginia Institute (1842–1845)Virginia Collegiate Institute (1845–1853)MottoPalmam Qui Meruit Ferat[1]Motto in EnglishLet him who earns the palm bear it.TypePrivate liberal arts collegeEstablished1842; 182 years ago (1842)Religious affiliationEvangelical Lutheran Church in AmericaAcademic affiliationsAnnapolis GroupEndowment$142.3 million (2020)[2]PresidentFrank Shushok ...

St Francis' Church, Melbourne

Church in Melbourne, AustraliaSt Francis' Catholic ChurchSt Francis' Church, Melbourne37°48′42″S 144°57′45″E / 37.8116°S 144.9624°E / -37.8116; 144.9624LocationMelbourneCountryAustraliaDenominationRoman CatholicReligious orderCongregation of the Blessed SacramentWebsitewww.stfrancismelbourne.comHistoryStatusChurch (1842 – 1848)Cathedral (1848 – 1868)Church {since 1868)Founded4 October 1841 (1841-10-04)Dedication...

Voyage of the Little Mermaid

Defunct live show at Disney's Hollywood Studios This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: Voyage of the Little Mermaid – news · newspapers · books · scholar · JSTOR (November 2015) (Learn how and when to remove this message) Voyage of the Little MermaidDisney's Hollywood StudiosAreaAnimation CourtyardC...

哥伦布 (密西西比州)

密西西比州哥伦布城市綽號：Possum Town哥伦布位于密西西比州的位置坐标：33°30′06″N 88°24′54″W / 33.501666666667°N 88.415°W / 33.501666666667; -88.415国家美國州密西西比州县朗兹县始建于1821年政府 • 市长罗伯特·史密斯 (民主党)面积 • 总计22.3 平方英里（57.8 平方公里） • 陸地21.4 平方英里（55.5 平方公里） • ...

Manhunt International 1997

Manhunt International 1997Tanggal24 Mei 1997TempatSingapuraPeserta38Finalis/Semifinalis10DebutBahamas, Gibraltar, Kolombia, Kosta Rika, Panama, Republik Makedonia, Ukraina, Uruguay, VenezuelaTidak tampilAustria, Britania Raya, El Salvador, Estonia, Fiji, Guatemala, Hong Kong, Hawaii, Nepal, Prancis, Republik Dominika, RumaniaTampil kembaliKanada, Irlandia, Spanyol, SwediaPemenangJason Erceg( Selandia Baru)Best National CostumeSandro Finoglio Speranza( Venezuela)Mr. TalentV...

Wayne M. Collins

American civil rights lawyer (1899–1974) Wayne M. CollinsBorn(1899-11-23)November 23, 1899[1]Sacramento, CaliforniaDiedJuly 16, 1974(1974-07-16) (aged 74)NationalityAmericanOccupationCivil Rights Attorney Wayne Mortimer Collins (November 23, 1899 – July 16, 1974) was a civil rights attorney who worked on cases related to the Japanese American evacuation and internment. Biography Personal life Collins was born in Sacramento, California, to Irish American parents, and was r...

Padre Paulo Ricardo

Priest Paul Richard at the Human Rights and Legislative Participation Commission, Brasília, 2016 This article is part of a series onConservatism in Brazil Ideologies Bolsonarism Coronelism Integralism Janismo Liberal conservatism Free Brazil Movement Monarchism Patrianovism Nationalism Populism Principles Agrarianism Anti-communism Authority Catholicism Family values Federalism Gun rights Law and order Militarism Patriotism Traditionalism History Brazil above everything, God above everyone B...

Iran Novin Party

1964–1975 ruling party of Iran Not to be confused with Iran-e-No Party. New Iran Party حزب ایران نوینFounderHassan Ali MansourFounded15 December 1963Dissolved2 March 1975Preceded byNationalists' Party[1]Merged intoRastakhiz Party[2]Worker wingWorker House (1967–75)IdeologyRoyalismPolitical positionRight-wingPolitics of IranPolitical partiesElections The Iran Novin Party (Persian: حزب ایران نوین, romanized: Ḥezb-e Īrān-e Novī...

Wilhelm von Brincken

German actor (1881-1946) Wilhelm von BrinckenBrincken in International Crime (1938)BornWilhelm Vaughn(1881-05-27)May 27, 1881Flensburg, Kingdom of Prussia, German EmpireDiedJanuary 18, 1946(1946-01-18) (aged 64)Los Angeles, California, U.S.Resting placeForest Lawn Memorial Park, Glendale, CaliforniaOther namesWilliam Vaughn, William Vaughan, William von BrinkenOccupationActorYears active1914–1944Spouses Alice M. Roedel (m. 1910; div. 1...

Methuen, Massachusetts

For other uses, see Methuen (disambiguation). City in Massachusetts, United StatesMethuen, MassachusettsCityMethuen City Hall FlagSealLocation in Essex County and the state of Massachusetts.Methuen, MassachusettsLocation in the United StatesCoordinates: 42°43′34″N 71°11′29″W / 42.72611°N 71.19139°W / 42.72611; -71.19139CountryUnited StatesStateMassachusettsCountyEssexSettled1642IncorporatedDecember 8, 1725Incorporated a city1917Named forSir Paul MethuenGove...

Dari Hati

/profillengkap.com/index.php/article/George Sykes (New Hampshire politician)

/profillengkap.com/index.php/article/Battle of Wilmington

/profillengkap.com/index.php/Genesis Mining

Proyek Z (proyek pengebom)

Gedalia Schorr

Jeruk, Bandar, Pacitan

2019 Toray Pan Pacific Open