I could be mistaken, but I expected errors here:
using TensorAlgebra: VectorSpace, Covector
V = VectorSpace(:V,Float64)
v = Vector(V,[1,2,3])
α = Covector(V,[1,2,3])
@show v(v)
@show α(α)
Or I expected something like an outer product e.g.
julia> Float64[1,2,3] * Float64[1,2,3]'
3×3 Array{Float64,2}:
1.0 2.0 3.0
2.0 4.0 6.0
3.0 6.0 9.0