Following some lines of joint work with A.S. Householder, the character and use of algebraic methods in the theory of norms is demonstrated. New results concerning norms with values in an Archimedian vector lattice (not necessarily being totally ordered) are given, in particular for the generalization of order unit norms, L-norms and M-norms. An example of application to operator norms is given concerning contraction properties of positive operators.
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