Halmos Lin Algebra Problem - clarification

I'm ploughing through Halmos Problem Book. He defines a vector space as a
commutative *group* V of vectors, a field F of scalars and a scalar
multiplication rule.

In Problem 21(4) he declares "Let F be R (the reals) and V the *set * Rplus
of all real positive numbers..." He then defines a peculiar addition rule in
V (a+b = ab) and a scalar multiplication rule ab = b exp a and invites the
reader to establish whether this rule makes V a vector space, by checking
through all the axioms. His answer is "yes", the addition and scalar
multiplication rules 'make' V a vector space.

But - how could V be a vector space, whatever the scalar multiplication
rule? Surely Rplus is not a group (no zero, no additive inverses)? Does
Halmos' vector addition rule make the set V a comutative group?

Would be grateful for a clarification.

Guy Corrigall


.

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