I think you're describing the evaluation map T(v, w) = w(v), which has type (1,1), rather than the inner product, which has type (2,0). The inner product lets y

I think you're describing the evaluation map T(v, w) = w(v), which has type (1,1), rather than the inner product, which has type (2,0). The inner product lets you "raise and lower indices" (i.e. convert between vectors and dual vectors), so you can basically pretend that it is the evaluation map.