*absolute*if for all , where is applied componentwise and

*monotone*if whenever componentwise.

Norms having these properties are also called

*Riesz*norms; the two conditions are equivalent for finite-dimensional spaces (see Horn and Johnson).

What about infinite-dimensional spaces? Does anyone have an example of a normed where the norm is satisfies one of the conditions but not the other? What about a function space?

For I think I have a proof that the two conditions are equivalent (by a handwavy appeal to Lebesgue's Dominated Convergence theorem). For function spaces, I suspect there's a counterexample. But this is all random 4am musing -- so please catch me if I'm disseminating lies!

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