Riesz norms

Let be a vector space endowed with a norm . A norm is called 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!