TL;DR
- Vector spaces are first class.
- Tensors are linear maps from a vector space to an underlying field t: V\to K
- Dual spaces are first class with
α = Covector(V,[1,2,3]) - Vectors are degree-1 tensors whose domain is the dual space, i.e. v: V^*\to K.
- Covectors are degree-1 tensors whose domain is the vector space, i.e. \alpha: V\to K.
- Partial evaluation of tensors is supported resulting in other tensors.
- No special indexing is required.
The second to last point is kind of neat. Given a tensor t\in V\otimes W for vector spaces V and W, we get 3 (5 really) maps for free:
- t: V^*\otimes W^*\to K,\quad\alpha\otimes\beta\mapsto t(\alpha\otimes\beta)\in K
- t: W^*\to V^*,\quad\beta\mapsto t(-,\beta)\in V^*
- t: V^*\to W^*,\quad\alpha\mapsto t(\alpha,-)\in W^*
julia> t(α⊗β)
500.0
julia> t(α,β)
500.0
julia> t(-,β)
3-element Tensor{Float64,1,V^*}:
30.0
70.0
110.0
julia> t(-,β) ∈ V
true
julia> t(α,-)
4-element Tensor{Float64,1,W^*}:
38.0
44.0
50.0
56.0
julia> t(α,-) ∈ W
true
julia> t(α,β) === t(-,β)(α) === t(α,-)(β)
true
Longer version
Hi everyone 
Many of you will remember the epic “Taking vector transposes seriously #4774”.
That was a heartening discussion that shows the Julia community does take this stuff seriously. As a result, we have a pretty awesome LinearAlgebra standard library.
However, as seriously as we took transposes, I still think we can do a little better. There is not a lot of room left in the design space to improve things because LinearAlgebra is already very good, but I think there is still some room. In my mind, this was highlighted in the constructive (also epic) discussion that took place with the PR to LinearAlgebra:
That discussion resulted in the creation of a new package: TensorCore.jl.
In an attempt to summarize my thoughts, I created the issue:
There was additional constructive discussions there. However, at some point, you just need to shut up and write some code. So that is what I did 
The result is TensorAlgebra.jl.
I am watching
with keen interest. The timeline for something like that would be v2.0 at the earliest so I think that would be an opportune time to possibly make some improvements to LinearAlgebra so that higher order tensor algorithms (of use in quantum computing, epidemiology, differential geometry, category theory, etc.) can be implemented more naturally.
Here is a quick walkthrough:
julia> V = VectorSpace(:V,Float64)
V
julia> v = Vector(V,[1,2,3])
3-element Tensor{Float64,1,V^*}:
1.0
2.0
3.0
julia> v ∈ V
true
julia> α = Covector(V,[1,2,3])
3-element Tensor{Float64,1,V}:
1.0
2.0
3.0
julia> α ∈ V^*
true
julia> α(v)
14.0
julia> v(α)
14.0
julia> TensorSpace(V,W)
V ⊗ W
julia> V⊗W
V ⊗ W
julia> t = Tensor((V⊗W)^*,[1 2 3 4;5 6 7 8;9 10 11 12])
3×4 Tensor{Float64,2,V^* ⊗ W^*}:
1.0 2.0 3.0 4.0
5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
julia> t ∈ V⊗W
true
julia> α
3-element Tensor{Float64,1,V}:
1.0
2.0
3.0
julia> β = Covector(W,[1,2,3,4])
4-element Tensor{Float64,1,W}:
1.0
2.0
3.0
4.0
julia> α⊗β
3×4 TensorProduct{Float64,2,Tuple{Tensor{Float64,1,V},Tensor{Float64,1,W}}}:
1.0 2.0 3.0 4.0
2.0 4.0 6.0 8.0
3.0 6.0 9.0 12.0
julia> α⊗β ∈ (V⊗W)^*
true
julia> t(α⊗β)
500.0
julia> t(α,β)
500.0
julia> t(-,β)
3-element Tensor{Float64,1,V^*}:
30.0
70.0
110.0
julia> t(α,-)
4-element Tensor{Float64,1,W^*}:
38.0
44.0
50.0
56.0
julia> t(-,β) ∈ V
true
julia> t(α,-) ∈ W
true
julia> t(α,β) === t(-,β)(α) === t(α,-)(β)
true
julia> t[2,3]
7.0
julia> (α⊗β)[2,3]
6.0
would help. I suspect you are being bitten by similar issues. I think something like an 