If I am in two dimensional space, the meaning I have for the span is the usual one from linear algebra. But I do not know what it means to say the roots in a root system, R, span the inner product space, E. For two reasons:
Look a the diagram of root space of B_2 (drawn in the two dimensional euclidean plane) there are 8 roots but we are in 2 dimensional space. The usual meaning of span would mean we have a linearly dependent set.
I can't reconcile the idea that roots can be linear functionals acting on the cartan subalgebra as an inner product vector space but then forming a subset of a vector space that does not act on the cartan subalgebra.
I am sure I am missing something simple.