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In mathematics, specifically inorder theory andfunctional analysis, afilter${\displaystyle \mathcal{F}}$ in anorder completevector lattice${\displaystyle X}$ isorder convergent if it contains anorder bounded subset (that is, a subset contained in an interval of the form${\displaystyle [a,b]:=\{x\in X:a\le x\text{ and }x\le b\}}$) and if$${\displaystyle sup\left\{infS:S\in \mathrm{OBound}(X)\cap \mathcal{F}\right\}=inf\left\{supS:S\in \mathrm{OBound}(X)\cap \mathcal{F}\right\},}$$ where${\displaystyle \mathrm{OBound}(X)}$ is the set of all order bounded subsets ofX, in which case this common value is called theorder limit of${\displaystyle \mathcal{F}}$ in${\displaystyle X.}$^[1]

Order convergence plays an important role in the theory ofvector lattices because the definition of order convergence does not depend on any topology.

A net${\displaystyle {\left(x_{\alpha }\right)}_{\alpha \in A}}$ in avector lattice${\displaystyle X}$ is said todecrease to${\displaystyle x_0\in X}$ if${\displaystyle \alpha \le \beta }$ implies${\displaystyle x_{\beta }\le x_{\alpha }}$ and${\displaystyle x_0=inf\left\{x_{\alpha }:\alpha \in A\right\}}$ in${\displaystyle X.}$ A net${\displaystyle {\left(x_{\alpha }\right)}_{\alpha \in A}}$ in a vector lattice${\displaystyle X}$ is said toorder-converge to${\displaystyle x_0\in X}$ if there is a net${\displaystyle {\left(y_{\alpha }\right)}_{\alpha \in A}}$ in${\displaystyle X}$ that decreases to${\displaystyle 0}$ and satisfies${\displaystyle \left|x_{\alpha }-x_0\right|\le y_{\alpha }}$ for all${\displaystyle \alpha \in A}$.^[2]

A linear map${\displaystyle T:X\to Y}$ between vector lattices is said to beorder continuous if whenever${\displaystyle {\left(x_{\alpha }\right)}_{\alpha \in A}}$ is a net in${\displaystyle X}$ that order-converges to${\displaystyle x_0}$ in${\displaystyle X,}$ then the net${\displaystyle {\left(T\left(x_{\alpha }\right)\right)}_{\alpha \in A}}$ order-converges to${\displaystyle T\left(x_0\right)}$ in${\displaystyle Y.}$${\displaystyle T}$ is said to be sequentially order continuous if whenever${\displaystyle {\left(x_n\right)}_{n\in \mathbb{N}}}$ is a sequence in${\displaystyle X}$ that order-converges to${\displaystyle x_0}$ in${\displaystyle X,}$then the sequence${\displaystyle {\left(T\left(x_n\right)\right)}_{n\in \mathbb{N}}}$ order-converges to${\displaystyle T\left(x_0\right)}$ in${\displaystyle Y.}$^[2]

In anorder completevector lattice${\displaystyle X}$ whose order isregular,${\displaystyle X}$ is ofminimal type if and only if every order convergent filter in${\displaystyle X}$ converges when${\displaystyle X}$ is endowed with theorder topology.^[1]

• Banach lattice – Banach space with a compatible structure of a lattice
• Fréchet lattice – Topological vector lattice
• Locally convex lattice
• Normed lattice
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

• Khaleelulla, S. M. (1982).Counterexamples in Topological Vector Spaces.Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York:Springer-Verlag.ISBN 978-3-540-11565-6.OCLC 8588370.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999).Topological Vector Spaces.GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.ISBN 978-1-4612-7155-0.OCLC 840278135.

• Functional analysis
• Order theory

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