isdual for AbstractTensorMaps

[Yue-Zhengyuan]

This PR adds a utility function isdual(x::AbstractTensorMap) that returns a Boolean vector indicating which spaces of a tensor map are dual spaces. For example:

x = randn(ℂ^2 ⊗ (ℂ^3)' ← (ℂ^1)' ⊗ ℂ^4)
isdual(x) # returns [0, 1, 0, 1] as a Boolean vector

It is mainly used to assert that arrows in a network are preserved, which I frequently need to do.

The name of the function can be changed if you feel that isdual must return a single Bool. I don't know how to meaningfully test it, either.

[Jutho]

I am not a great fan of calling isdual on a tensor, since that might have other connotations as well. One way forward that I could see is the following, but I would certainly like the opinion of @lkdvos .

Firstly, we could have a method:

spaces(t::AbstractTensorMap) = (codomain(t)..., map(dual, domain(t))...)

and then you can simply use broadcasting to write isdual.(spaces(t)). Would that be a sufficient shorthand?

[Jutho]

Also, in relation to this and #430 , while map( f, ::ProductSpace) now indeed returns a tuple, broadcasting f.(::ProductSpace) still returns a vector, so we probably should also fix the broadcasting behavior of ProductSpace objects.

[Yue-Zhengyuan]

you can simply use broadcasting to write isdual.(spaces(t)). Would that be a sufficient shorthand?

This will be convenient enough for me 👍. Or maybe even better, instead of write a new spaces, we can define how to iterate over TensorMapSpace.

broadcasting f.(::ProductSpace) still returns a vector

It appears to have a simple solution: explicitly define

Base.Broadcast.broadcastable(P::ProductSpace) = P.spaces

Otherwise it defaults to collect(x), resulting in a vector.

[lkdvos]

Summarizing my earlier comment, I would like something like this: isdual.(codomain(t)), isdual.(domain(t)) (with appropriate negations), but I don't have strong opinions on adding a utility function to replace that if this is really desired (if we can come up with a good name).

The following also would make sense to me: in Base we often have the pattern size(A, i) = size(A)[i], so if we have space(t, i) = space(t)[i] (which I think we already do?), I would be fine making isdual.(space(t)) broadcast over the codomain, dual(domain) spaces in a linear way.

[Yue-Zhengyuan]

if we have space(t, i) = space(t)[i] (which I think we already do?), I would be fine making isdual.(space(t)) broadcast over the codomain, dual(domain) spaces in a linear way.

We do have space(t, i) = space(t)[i]. The only thing preventing isdual.(space(t)) is the lack of Base.iterate for HomSpace. Also be careful that dual(domain) will reverse the leg order in the domain, which I don't want in isdual.(space(t)).

[Jutho]

Given that we already support getindex on HomSpace, I would be supportive of both broadcasting and iteration support for ProductSpace and HomSpace.

Can you implement the necessary functionality for this in the current PR?