Surds as a vector space, but really an analysis question I think

[Office_Shredder]

I guess this is a bit of an interdisciplinary post... a friend of mine was looking at surds as a way of making a vector space with a basis as the prime numbers. Here's the construction:

V = {Surds} = {q^r|q is rational and positive, r is rational}

The field that V is over is Q

And we defined vector space multiplication and addition (* and +)as follows for vectors p,q and scalars r,s

p+q:= pq
r*p:= p^r

and this forms a vector space with the "zero element" as 1 and multiplicative inverses in Q as the additive inverses in V, and the prime numbers form a basis

Now, the motivation here is to study the basis... the first thought was to perform Graham Schmidt orthogonalization (at least on finite subsets). Ok, so we need a norm. Here's what we can glean:

|.|:V->R
|r*p| = |p^r| = |r|*|p| = r*|p|
and
|1| = 0
That looks a lot like a logarithm to me actually... are there any other functions that have this property? The problem with having a logarithm for the norm is that to form an orthonormal basis, you need the norms of your basis elements to lie in the field, which in this case is Q, and I don't think there exists a base b such that log_bq^r is in Q for all q^r in V

 

[lurflurf]

It depends what you mean by like a logarithm.
Use the "obvious" norm
||v||=||(2^a)(3^b)(5^c)(7^d)...||=||(a,b,c,d,...||=sqrt[a^2+b^2+c^2+d^2+...]