Order convergence

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

A linear T : X → Y between vector lattices is said to be order continuous if whenever ${\displaystyle \left(x_{\alpha }\right)_{\alpha \in A}}$ is a net in X that order-converges to x₀ in X, then the net ${\displaystyle \left(T\left(x_{\alpha }\right)\right)_{\alpha \in A}}$ order-converges to T(x₀) in Y. T is said to be sequentially order continuous if whenever ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ is a sequence in X that order-converges to x₀ in X, then the sequence ${\displaystyle \left(T\left(x_{n}\right)\right)_{n\in \mathbb {N} }}$ order-converges to T(x₀) in Y.[2]

In n order complete vector lattice X whose order is regular, X is of minimal type if and only if every order convergent filter in X converges when X is endowed with the order topology.[1]

• Banach lattice
• Fréchet lattice
• Locally convex lattice
• Normed lattice
• Vector lattice

• Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. 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. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.