Consider the space of real functions $$ X:=\bigg\{e^{\frac{\log^2(s)}{\log(x)}}:x\in(0,1),s\in(0,1]\bigg\}. $$
The map $\rho:e^{\frac{\log^2(s)}{\log(x)}}\mapsto s$ is a way to associate a magnitude to each function in the space. So I guess this is the norm of the function space?
I read that the norm is a way to assign a positive real number to each function in the function space, so I think it is a norm.
How does the norm I wrote tell me anything about the function space?
So from $X$'s vector space structure, if $s$ is real, then it's a 1-dimensional $\Bbb R$ vector space, and is the open real unit interval I think.
And I think the norm, $\rho$ gives the same information as the usual Euclidean metric on the open unit interval?