|
|
Guy Corrigall wrote:
> 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?
Yes. In this case the zero element is the real number 1. If a>0, it's
inverse is 1/a.
|
|