View source for Supremum

==English==

===Etymology===
Borrowed from {{bor|en|la|supremum}}.

===Pronunciation===
* {{audio|en|LL-Q1860 (eng)-Flame, not lame-supremum.wav|a=US}}

===Noun===
{{en-noun|s|suprema}}

# {{wikipedia}}{{lb|en|set theory}} {{lb|en|real analysis}}: Given a [[subset]] ''X'' of '''R''', the smallest [[real number]] that is ≥ every element of ''X''; {{lb|en|order theory}}: given a [[subset]] ''X'' of a [[partially ordered set]] ''P'' (with [[partial order]] ≤), the [[least]] element ''y'' of ''P'' such that every element of ''X'' is ≤ ''y''.
#* {{quote-book|en|year=2006|author=Charalambos D. Aliprantis; Kim C. Border|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3rd|pageurl=https://books.google.com.au/books?id=4hIq6ExH7NoC&pg=PA8&dq=%22supremum%22%7C%22suprema%22&hl=en&sa=X&ved=0ahUKEwiKnK-l4vXbAhXLFJQKHRD1CPEQ6AEIdjAQ#v=onepage&q=%22supremum%22%7C%22suprema%22&f=false|page=8|publisher=Springer
|passage=A '''sublattice''' of a lattice is a subset that is closed under pairwise infima and '''suprema'''.}}
#* {{quote-book|en|year=2010|author=James S. Howland|title=Basic Real Analysis|publisher={{w|Jones & Bartlett Learning|Jones & Bartlett Publishers}}|pageurl=https://books.google.com.au/books?id=zyNXBggk8GEC&pg=PA9&dq=%22supremum%22%7C%22suprema%22&hl=en&sa=X&ved=0ahUKEwiKnK-l4vXbAhXLFJQKHRD1CPEQ6AEIMTAB#v=onepage&q=%22supremum%22%7C%22suprema%22&f=false|page=9
|passage=The best way to describe the '''supremum''' of ''S'' is to say that it wants to be the greatest element of ''S''. In fact, if ''S'' has a greatest element, then that element is the '''supremum'''.}}
#* {{quote-book|en|year=2011|author=Andreas Löhne|title=Vector Optimization with Infimum and '''Supremum'''|pageurl=https://books.google.com.au/books?id=gvbcBIFp3xEC&pg=PR7&dq=%22supremum%22%7C%22suprema%22&hl=en&sa=X&ved=0ahUKEwiKnK-l4vXbAhXLFJQKHRD1CPEQ6AEINzAC#v=onepage&q=%22supremum%22%7C%22suprema%22&f=false|page=vii|publisher=Springer
|passage=The key to an approach to vector optimization based on infimum and '''supremum''' is to consider set-based objective functions and to extend the partial ordering of the original objective space to a suitable subspace of the power set. In this new space the infimum and '''supremum''' exist under the usual assumptions.}}

====Usage notes====
* Commonly denoted sup(''X'').
* The supremum of ''X'' may not exist, and, if it does, may not be an element of ''X''.
* {{sense|order theory}}
** Formally: Let <math>S =\{ t : t \in P : \forall x\in X, x \le t \}</math> be the set of [[upper]] [[bound]]s of ''X''. Then sup(''X''), if it exists, is the element <math>s\in S: \forall y\in S, s \le y</math>.
** The concept of ''supremum'' is closely related to the function [[∨]] (called {{m|en|join}}). The supremum of two elements, denoted <math>\sup\{x,y\}</math> can also be written <math>x\lor y</math>. The supremum of a set may be denoted <math>\sup(X)</math> or <math>\bigvee X</math>.

====Synonyms====
* {{sense|element of a set greater than or equal to all members of a given subset}} {{l|en|least upper bound}}, {{l|en|LUB}}, {{l|en|sup}}

====Coordinate terms====
* {{l|en|infimum}}

====Derived terms====
{{col|en|essential supremum}}

====Translations====
{{trans-top|element of a set greater than or equal to all members of a given subset}}
* Czech: {{t|cs|supremum|n}}
* Finnish: {{t|fi|[[pienin]] [[yläraja]]}}, {{t+|fi|supremum}}
* French: {{t+|fr|supremum|m}}
* German: {{t+|de|Supremum|n}}
* Hebrew: {{t|he|סופרמום|m}}, {{t|he|חסם עליון|m}}
* Japanese: {{t+|ja|上限|tr=じょうげん, jōgen|sc=Jpan}}
* Polish: {{t+|pl|supremum|n}}, {{t+|pl|kres górny|m}}
* Spanish: {{t+|es|supremo|m}}
* Swedish: {{t+|sv|supremum}}
{{trans-bottom}}

===See also===
* {{l|en|maximum}}

===Further reading===
* {{pedia|Infimum and supremum}}
* {{pedia|Join and meet}}
* {{pedia|Lattice (order)}}
* {{pedia|Least-upper-bound property}}
* {{pedia|Upper and lower bounds}}

===Anagrams===
* {{anagrams|en|a=emmprsuu|supermum}}

{{C|en|Algebra|Mathematical analysis}}

==Czech==

===Pronunciation===
* {{cs-IPA|[rém]}}

===Noun===
{{cs-noun|n}}

# {{lb|cs|mathematics}} {{l|en|supremum}}
#: {{ant|cs|infimum}}

====Declension====
{{cs-ndecl|n.foreign}}

===Further reading===
* {{R:cs:ASCS}}

==Finnish==

===Etymology===
{{lbor|fi|la|suprēmum}}.

===Pronunciation===
{{fi-pronunciation|supre(:)mum}}

===Noun===
{{fi-noun}}

# {{lb|fi|mathematics}} [[#English|supremum]]

====Declension====
{{fi-decl-risti|supremum|||a|i=0}}

====Synonyms====
* {{l|fi|[[pienin]] [[yläraja]]}}

====Antonyms====
* {{l|fi|infimum}}

==Latin==

===Adjective===
{{head|la|adjective form|head=suprēmum}}

# {{inflection of|la|suprēmus||nom//acc//voc|n|s|;|acc|m|s}}

===References===
* {{R:Elementary Lewis}}
* {{R:Gaffiot}}
* {{R:M&A}}

==Swedish==

===Noun===
{{sv-noun|n}}

# {{lb|sv|mathematics}} supremum

====Declension====
{{sv-infl-noun-n-zero}}