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Banach Lattices and Positive Operators (Grundlehren der mathematischen
Wissenschaften, 215)
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Vector lattices-also called Riesz spaces, K-lineals, or linear lattices-were first considered by F.
Riesz, L. Kantorovic, and H. Freudenthal in the middle nineteen thirties thus their early theory
dates back almost as far as the beginning of the systematic investigation of Banach spaces.
Schools of research on vector lattices were subsequently founded in the Soviet Union (Kantorovic,
Judin, Pinsker, Vulikh) and in Japan (Nakano, Ogasawara, Yosida) other important contri­ butions
came from the United States (G. Birkhoff, Kakutani, M. H. Stone). L. Kantorovic and his school first
recognized the importance of studying vector lattices in connection with Banach's theory of
normed vector spaces they investigated normed vector lattices as well as order-related linear
operators between such vector lattices. (Cf. Kantorovic-Vulikh-Pinsker [1950] and Vulikh [1967].)
However, in the years following that early period, functional analysis and vector lattice theory
began drifting more and more apart it is my impression that 'linear order theory' could not quite
keep pace with the rapid development of general functional analysis and thus developed into a
theory largely existing for its own sake, even though it had interesting and beautiful applications
here and there. em em
Vector lattices-also called Riesz spaces, K-lineals, or linear lattices-were first considered by F.
Riesz, L. Kantorovic, and H. Freudenthal in the middle nineteen thirties thus their early theory
dates back almost as far as the beginning of the systematic investigation of Banach spaces.
Schools of research on vector lattices were subsequently founded in the Soviet Union (Kantorovic,
Judin, Pinsker, Vulikh) and in Japan (Nakano, Ogasawara, Yosida) other important contri­ butions
came from the United States (G. Birkhoff, Kakutani, M. H. Stone). L. Kantorovic and his school first
recognized the importance of studying vector lattices in connection with Banach's theory of
normed vector spaces they investigated normed vector lattices as well as order-related linear
operators between such vector lattices. (Cf. Kantorovic-Vulikh-Pinsker [1950] and Vulikh [1967].)
However, in the years following that early period, functional analysis and vector lattice theory
began drifting more and more apart it is my impression that 'linear order theory' could not quite
keep pace with the rapid development of general functional analysis and thus developed into a
theory largely existing for its own sake, even though it had interesting and beautiful applications
here and there. em em