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In [[mathematics]], more specifically in [[functional analysis]], a subset <math>T</math> of a [[topological vector space]] <math>X</math> This condition arises frequently in many theorems of functional analysis. ...

In the theory of [[orthogonal functions]], '''Lauricella's theorem''' provides ..._k\}</math> [[Convergence_of_random_variables#Convergence_in_mean|converge in the mean]] to that function. ...

...|mr=2009|pages=556–583|series=Second Series|title=On regularly convex sets in the space conjugate to a Banach space|volume=41|year=1940|issue=3 |jstor=19 Both of the following theorems are referred to as the Krein-Smulian Theorem. ...

...ly continuous operator|first1=Victor I.|last1=Lomonosov|journal=Functional Analysis and Its Applications|volume=7|pages=213–214|date=1973|issue=3 |doi=10.1007/ ...if <math>T(M)\subset M</math>, i.e. <math>Tx\in M</math> for every <math>x\in M</math>. ...

...everywhere of operator averages |journal=Journal of Rational Mechanics and Analysis |volume=5 |year=1956 |pages=129–178 |mr=77090 }}.</ref> exists almost everywhere for all <math display="inline">f\in L^1</math>. ...

In [[mathematics]], '''Dieudonné's theorem''', named after [[Jean Dieudonné]], ...escu">{{cite book |last=Zălinescu |first=Constantin |title=Convex analysis in general vector spaces |publisher=World Scientific Publishing&nbsp;Co.,&nbsp ...

...] and [[Norman George Meyers]], states that [[smooth functions]] are dense in the [[Sobolev space]] <math>W^{k,p}(\Omega)</math> ...n ended much confusion about the relationship of these spaces that existed in the literature before that time. It is surprising that this elementary resu ...

...rem''' is a result in the field of [[functional analysis]]. It states that in a [[unital algebra|unital]] [[C*-algebra]], the closure of the [[convex hul ...A. Belfi | title = Characterizations of C*-Algebras: The Gelfand–Naimark Theorems | publisher = Marcel Dekker | location = New York | year = 1986 | isbn = 0- ...

...ection]] of every [[decreasing sequence]] of [[Ball (mathematics)|balls]] (in the sense of the metric induced by the absolute value) is nonempty:<ref>{{C :<math>B_1\supseteq B_2\supseteq \cdots \Rightarrow\bigcap_{n\in {\mathbf N}} B_n\neq \empty.</math> ...

...</math> and <math> w \in V </math>) whenever <math> \displaystyle \lim_{i \in I} u_{i} v = v </math>. The theorem was introduced by {{harvs|txt|authorlin |title = Factorization in group algebras ...

In [[functional analysis]], the '''Borel graph theorem''' is generalization of the [[closed graph th ...id for [[linear map]]s defined on and valued in most spaces encountered in analysis.{{sfn|Trèves|2006|p=549}} ...

...istinct from) the [[Weak topology|weak convergence]] in [[Banach space]]s. In [[Hilbert space]], Delta-convergence and weak convergence coincide. For a g ...d by Teck-Cheong Lim,<ref name="Lim">T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. '''60''' (1976), 179–182.</ref> and, soon after, un ...

...ogical vector space]] has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups. ...<math>(0,1)</math> and <math>T</math> in <math>S</math>. Then the mappings in <math>S</math> share a fixed point.{{sfn | Conway | 1990 | pp=151–152}} ...

....<ref>{{cite book|last1=Aliprantis|last2=Border|title=Infinite-dimensional analysis. A hitchhiker's guide.|date=2006}}</ref><ref>{{cite book|first=Alexander S. ...st3=José |date=2010 |title=Measurability and Selections of Multi-Functions in Banach Spaces ...

...med after [[Errett Bishop]] and [[Robert Phelps]], who published its proof in 1961.<ref name="BishopPhelps1961">{{cite journal|last1=Bishop|first1=Errett ...ere exists some <math>b_0 \in B</math> such that <math>|f(b_0)| = \sup_{b \in B} |f(b)|</math>) ...

...ntrol]], and [[mathematical economics]].<ref>{{cite book|title=Fixed Point Theorems with Applications to Economics and Game Theory|last=Border|first=Kim C.|pub : <math>\forall x \in X: \,\,\, f(x) \in F(x) \,.</math> ...

...y order. It is a special case of the [[Burkholder-Davis-Gundy inequality]] in the case of discrete-time martingales. ...fonctions indépendantes. ''Fund. Math.'', 28:60&ndash;90, 1937. Reprinted in Józef Marcinkiewicz, ''Collected papers'', edited by Antoni Zygmund, Panstw ...

In the [[mathematics|mathematical]] theory of [[Banach space]]s, the '''closed The theorem was proved by [[Stefan Banach]] in his [[1932]] ''[[Théorie des opérations linéaires]]''. ...

...'' (named after [[Frigyes Riesz]]) is an important theorem in [[functional analysis]] that states that a [[Hausdorff space|Hausdorff]] [[topological vector spa The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned. ...

...ection theorem''' is a [[theorem]] proved by [[Wilhelm Wirtinger]] in 1932 in connection with some problems of [[approximation theory]]. This theorem giv }}</ref> contains the following theorem presented also in [[Joseph L. Walsh]]'s well-known monograph ...