A norm $||.||$ on a Riesz space is said to be lattice norm whenever $|x|\leq |y|$ implies $||x||\leq ||y||$. Do you know any example which is not a lattice norm?
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Where is this definition of lattice norm from? What is your source for this? Also is $|x| := (x) \vee (-x)$? – Aug 10 '15 at 12:41
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Take the Sobolev space $H^1(0,1)$. For example, you could choose $y \equiv 1$ and $x(t) = \sin(n \, t)$ for $n$ large enough.
gerw
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Hi, do you know a good source for where this idea of lattice norm is defined. Also is $|x| := (x) \vee (-x)$? Thanks. – Aug 10 '15 at 12:43
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You can find it, e.g., in Schaefer's "Banach Lattices and Positive Operators", p.5, (8). – gerw Aug 10 '15 at 13:10