Merge pull request #72 from lkdvos/spacetypes

Refactor: abstract spacetypes to traits

- Removes `InnerProductSpace` and `EuclideanSpace` as abstract types.
- Adds `InnerProductStyle` as a trait based mechanism to query if a space has an inner product.
- Implements `NoInnerProduct`, `HasInnerProduct` and `EuclideanInnerProduct` as traits

Author: lkdvos

.gitignore

@@ -5,3 +5,4 @@
 *-old
 __pycache__
 .ipynb*
+Manifest.toml

docs/src/lib/spaces.md

@@ -11,8 +11,6 @@ Field
 VectorSpace
 ElementarySpace
 GeneralSpace
-InnerProductSpace
-EuclideanSpace
 CartesianSpace
 ComplexSpace
 GradedSpace

docs/src/man/sectors.md

@@ -647,9 +647,9 @@ as illustrated below.
 We have introduced `Sector` subtypes as a way to label the irreps or sectors in the
 decomposition ``V = ⨁_a ℂ^{n_a} ⊗ R_{a}``. To actually represent such spaces, we now also
 introduce a corresponding type `GradedSpace`, which is a subtype of
-`EuclideanSpace{ℂ}`, i.e.
+`ElementarySpace{ℂ}`, i.e.
 ```julia
-struct GradedSpace{I<:Sector, D} <: EuclideanSpace{ℂ}
+struct GradedSpace{I<:Sector, D} <: ElementarySpace{ℂ}
     dims::D
     dual::Bool
 end

docs/src/man/spaces.md

@@ -103,40 +103,35 @@ struct GeneralSpace{𝕜} <: ElementarySpace{𝕜}
 end
 ```
 
-We furthermore define the abstract type
+We furthermore define the trait types
 ```julia
-abstract type InnerProductSpace{𝕜} <: ElementarySpace{𝕜} end
+abstract type InnerProductStyle end
+struct NoInnerProduct <: InnerProductStyle end
+abstract type HasInnerProduct <: InnerProductStyle end
+struct EuclideanProduct <: HasInnerProduct end
 ```
-to contain all vector spaces `V` which have an inner product and thus a canonical mapping
-from `dual(V)` to `V` (for `𝕜 ⊆ ℝ`) or from `dual(V)` to `conj(V)` (otherwise). This
-mapping is provided by the metric, but no further support for working with metrics is
+to denote for a vector space `V` whether it has an inner product and thus a canonical
+mapping from `dual(V)` to `V` (for `𝕜 ⊆ ℝ`) or from `dual(V)` to `conj(V)` (otherwise).
+This mapping is provided by the metric, but no further support for working with metrics is
 currently implemented.
 
-Finally there is
-```julia
-abstract type EuclideanSpace{𝕜} <: InnerProductSpace{𝕜} end
-```
-to contain all spaces `V` with a standard Euclidean inner product (i.e. where the metric is
-the identity). These spaces have the natural isomorphisms `dual(V) == V` (for `𝕜 == ℝ`)
-or `dual(V) == conj(V)` (for ` 𝕜 == ℂ`). In the language of the previous section on
-[categories](@ref s_categories), this subtype represents
-[dagger or unitary categories](@ref ss_adjoints), and support an `adjoint` operation. In
-particular, we have two concrete types
+The `EuclideanProduct` has the natural isomorphisms `dual(V) == V` (for `𝕜 == ℝ`)
+or `dual(V) == conj(V)` (for ` 𝕜 == ℂ`).
+In the language of the previous section on [categories](@ref s_categories), this trait represents [dagger or unitary categories](@ref ss_adjoints), and these vector spaces support an `adjoint` operation.
+
+In particular, the two concrete types
 ```julia
-struct CartesianSpace <: EuclideanSpace{ℝ}
+struct CartesianSpace <: ElementarySpace{ℝ}
     d::Int
 end
-struct ComplexSpace <: EuclideanSpace{ℂ}
+struct ComplexSpace <: ElementarySpace{ℂ}
   d::Int
   dual::Bool
 end
 ```
-to represent the Euclidean spaces $ℝ^d$ or $ℂ^d$ without further inner structure. They can
-be created using the syntax `CartesianSpace(d) == ℝ^d == ℝ[d]` and
-`ComplexSpace(d) == ℂ^d == ℂ[d]`, or
-`ComplexSpace(d, true) == ComplexSpace(d; dual = true) == (ℂ^d)' == ℂ[d]'` for the
-dual space of the latter. Note that the brackets are required because of the precedence
-rules, since `d' == d` for `d::Integer`.
+represent the Euclidean spaces $ℝ^d$ or $ℂ^d$ without further inner structure.
+They can be created using the syntax `CartesianSpace(d) == ℝ^d == ℝ[d]` and `ComplexSpace(d) == ℂ^d == ℂ[d]`, or `ComplexSpace(d, true) == ComplexSpace(d; dual = true) == (ℂ^d)' == ℂ[d]'` for the dual space of the latter.
+Note that the brackets are required because of the precedence rules, since `d' == d` for `d::Integer`.
 
 Some examples:
 ```@repl tensorkit
@@ -149,12 +144,15 @@ dual(ℂ^5) == (ℂ^5)' == conj(ℂ^5) == ComplexSpace(5; dual = true)
 typeof(ℝ^3)
 spacetype(ℝ^3)
 spacetype(ℝ[])
+InnerProductStyle(ℝ^3)
+InnerProductStyle(ℂ^5)
 ```
+
 Note that `ℝ[]` and `ℂ[]` are synonyms for `CartesianSpace` and `ComplexSpace` respectively,
 such that yet another syntax is e.g. `ℂ[](d)`. This is not very useful in itself, and is
 motivated by its generalization to `GradedSpace`. We refer to the subsection on
 [graded spaces](@ref s_rep) on the [next page](@ref s_sectorsrepfusion) for further
-information about `GradedSpace`, which is another subtype of `EuclideanSpace{ℂ}`
+information about `GradedSpace`, which is another subtype of `ElementarySpace{ℂ}`
 with an inner structure corresponding to the irreducible representations of a group, or more
 generally, the simple objects of a fusion category.
 
@@ -279,7 +277,7 @@ For completeness, we also export the strict comparison operators `≺` and `≻`
 ```
 However, as we expect these to be less commonly used, no ASCII alternative is provided.
 
-In the context of `spacetype(V) <: EuclideanSpace`, `V1 ≾ V2` implies that there exists
+In the context of `InnerProductStyle(V) <: EuclideanProduct`, `V1 ≾ V2` implies that there exists
 isometries ``W:V1 → V2`` such that ``W^† ∘ W = \mathrm{id}_{V1}``, while `V1 ≅ V2` implies
 that there exist unitaries ``U:V1→V2`` such that ``U^† ∘ U = \mathrm{id}_{V1}`` and
 ``U ∘ U^† = \mathrm{id}_{V2}``.

docs/src/man/tensors.md

@@ -391,17 +391,18 @@ can use the method `u = isomorphism([A::Type{<:DenseMatrix}, ] codomain, domain)
 will explicitly check that the domain and codomain are isomorphic, and return an error
 otherwise. Again, an optional first argument can be given to specify the specific type of
 `DenseMatrix` that is currently used to store the rather trivial data of this tensor. If
-`spacetype(u) <: EuclideanSpace`, the same result can be obtained with the method `u =
-unitary([A::Type{<:DenseMatrix}, ] codomain, domain)`. Note that reversing the domain and
-codomain yields the inverse morphism, which in the case of `EuclideanSpace` coincides with
-the adjoint morphism, i.e. `isomorphism(A, domain, codomain) == adjoint(u) == inv(u)`, where
-`inv` and `adjoint` will be further discussed [below](@ref ss_tensor_linalg). Finally, if
-two spaces `V1` and `V2` are such that `V2` can be embedded in `V1`, i.e. there exists an
-inclusion with a left inverse, and furthermore they represent tensor products of some
-`EuclideanSpace`, the function `w = isometry([A::Type{<:DenseMatrix}, ], V1, V2)` creates
-one specific isometric embedding, such that `adjoint(w)*w == id(V2)` and `w*adjoint(w)` is
-some hermitian idempotent (a.k.a. orthogonal projector) acting on `V1`. An error will be
-thrown if such a map cannot be constructed for the given domain and codomain.
+`InnerProductStyle(u) <: EuclideanProduct`, the same result can be obtained with the method
+`u = unitary([A::Type{<:DenseMatrix}, ] codomain, domain)`. Note that reversing the domain
+and codomain yields the inverse morphism, which in the case of `EuclideanProduct` coincides
+with the adjoint morphism, i.e. `isomorphism(A, domain, codomain) == adjoint(u) == inv(u)`,
+where `inv` and `adjoint` will be further discussed [below](@ref ss_tensor_linalg).
+Finally, if two spaces `V1` and `V2` are such that `V2` can be embedded in `V1`, i.e. there
+exists an inclusion with a left inverse, and furthermore they represent tensor products of
+some `ElementarySpace` with `EuclideanProduct`, the function
+`w = isometry([A::Type{<:DenseMatrix}, ], V1, V2)` creates one specific isometric
+embedding, such that `adjoint(w) * w == id(V2)` and `w * adjoint(w)` is some hermitian
+idempotent (a.k.a. orthogonal projector) acting on `V1`. An error will be thrown if such a
+map cannot be constructed for the given domain and codomain.
 
 Let's conclude this section with some examples with `GradedSpace`.
 ```@repl tensors
@@ -575,7 +576,7 @@ can only exist if the domain and codomain are isomorphic, which can e.g. be chec
 `t2`, we can use the syntax `t1\t2` or `t2/t1`. However, this syntax also accepts instances
 `t1` whose domain and codomain are not isomorphic, and then amounts to `pinv(t1)`, the
 Moore-Penrose pseudoinverse. This, however, is only really justified as minimizing the
-least squares problem if `spacetype(t) <: EuclideanSpace`.
+least squares problem if `InnerProductStyle(t) <: EuclideanProduct`.
 
 `AbstractTensorMap` instances behave themselves as vectors (i.e. they are `𝕜`-linear) and
 so they can be multiplied by scalars and, if they live in the same space, i.e. have the same
@@ -588,30 +589,29 @@ operations, TensorKit.jl reexports a number of efficient in-place methods from
 `lmul!` and `rmul!` (for `y ← α*y` and `y ← y*α`, which is typically the same) and `mul!`,
 which can also be used for out-of-place scalar multiplication `y ← α*x`.
 
-For `t::AbstractTensorMap{S}` where `S<:EuclideanSpace`, henceforth referred to as
-a `(Abstract)EuclideanTensorMap`, we can compute `norm(t)`, and for two such instances, the
-inner product `dot(t1, t2)`, provided `t1` and `t2` have the same domain and codomain.
-Furthermore, there is `normalize(t)` and `normalize!(t)` to return a scaled version of `t`
-with unit norm. These operations should also exist for `S<:InnerProductSpace`, but requires
-an interface for defining a custom inner product in these spaces. Currently, there is no
-concrete subtype of `InnerProductSpace` that is not a subtype of `EuclideanSpace`. In
-particular, `CartesianSpace`, `ComplexSpace` and `GradedSpace` are all subtypes
-of `EuclideanSpace`.
-
-With instances `t::AbstractEuclideanTensorMap` there is associated an adjoint operation,
-given by `adjoint(t)` or simply `t'`, such that `domain(t') == codomain(t)` and
-`codomain(t') == domain(t)`. Note that for an instance `t::TensorMap{S,N₁,N₂}`, `t'` is
-simply stored in a wrapper called `AdjointTensorMap{S,N₂,N₁}`, which is another subtype of
-`AbstractTensorMap`. This should be mostly unvisible to the user, as all methods should work
-for this type as well. It can be hard to reason about the index order of `t'`, i.e. index
-`i` of `t` appears in `t'` at index position `j = TensorKit.adjointtensorindex(t, i)`,
-where the latter method is typically not necessary and hence unexported. There is also a
-plural `TensorKit.adjointtensorindices` to convert multiple indices at once. Note that,
-because the adjoint interchanges domain and codomain, we have
-`space(t', j) == space(t, i)'`.
+For `t::AbstractTensorMap{S}` where `InnerProductStyle(S) <: EuclideanProduct`, we can
+compute `norm(t)`, and for two such instances, the inner product `dot(t1, t2)`, provided
+`t1` and `t2` have the same domain and codomain. Furthermore, there is `normalize(t)` and
+`normalize!(t)` to return a scaled version of `t` with unit norm. These operations should
+also exist for  `InnerProductStyle(S) <: HasInnerProduct`, but require an interface for
+defining a custom inner product in these spaces. Currently, there is no concrete subtype of
+`HasInnerProduct` that is not an `EuclideanProduct`. In particular, `CartesianSpace`,
+`ComplexSpace` and `GradedSpace` all have `InnerProductStyle(V) <: EuclideanProduct`.
+
+With tensors that have `InnerProductStyle(t) <: EuclideanProduct` there is associated an
+adjoint operation, given by `adjoint(t)` or simply `t'`, such that
+`domain(t') == codomain(t)` and `codomain(t') == domain(t)`. Note that for an instance
+`t::TensorMap{S,N₁,N₂}`, `t'` is simply stored in a wrapper called
+`AdjointTensorMap{S,N₂,N₁}`, which is another subtype of `AbstractTensorMap`. This should
+be mostly unvisible to the user, as all methods should work for this type as well. It can
+be hard to reason about the index order of `t'`, i.e. index `i` of `t` appears in `t'` at
+index position `j = TensorKit.adjointtensorindex(t, i)`, where the latter method is
+typically not necessary and hence unexported. There is also a plural
+`TensorKit.adjointtensorindices` to convert multiple indices at once. Note that, because
+the adjoint interchanges domain and codomain, we have `space(t', j) == space(t, i)'`.
 
 `AbstractTensorMap` instances can furthermore be tested for exact (`t1 == t2`) or
-approximate (`t1 ≈ t2`) equality, though the latter requires `norm` can be computed.
+approximate (`t1 ≈ t2`) equality, though the latter requires that `norm` can be computed.
 
 When tensor map instances are endomorphisms, i.e. they have the same domain and codomain,
 there is a multiplicative identity which can be obtained as `one(t)` or `one!(t)`, where the
@@ -798,8 +798,8 @@ inverse, to split a given index into two or more indices. For a plain tensor (i.
 data. However, this represents only one possibility, as there is no canonically unique way
 to embed the tensor product of two spaces `V₁ ⊗ V₂` in a new space `V = fuse(V₁⊗V₂)`. Such a
 mapping can always be accompagnied by a basis transform. However, one particular choice is
-created by the function `isomorphism`, or for `EuclideanSpace` spaces, `unitary`. Hence, we
-can join or fuse two indices of a tensor by first constructing
+created by the function `isomorphism`, or for `EuclideanProduct` spaces, `unitary`.
+Hence, we can join or fuse two indices of a tensor by first constructing
 `u = unitary(fuse(space(t, i) ⊗ space(t, j)), space(t, i) ⊗ space(t, j))` and then
 contracting this map with indices `i` and `j` of `t`, as explained in the section on
 [contracting tensors](@ref ss_tensor_contraction). Note, however, that a typical algorithm
@@ -851,7 +851,7 @@ U, Σ, Vʰ, ϵ = tsvd(t; trunc = notrunc(), p::Real = 2,
 ```
 
 This computes a (possibly truncated) singular value decomposition of
-`t::TensorMap{S,N₁,N₂}` (with `S<:EuclideanSpace`), such that
+`t::TensorMap{S,N₁,N₂}` (with `InnerProductStyle(t)<:EuclideanProduct`), such that
 `norm(t - U*Σ*Vʰ) ≈ ϵ`, where `U::TensorMap{S,N₁,1}`, `S::TensorMap{S,1,1}`,
 `Vʰ::TensorMap{S,1,N₂}` and `ϵ::Real`. `U` is an isometry, i.e. `U'*U` approximates the
 identity, whereas `U*U'` is an idempotent (squares to itself). The same holds for

docs/src/man/tutorial.md

@@ -28,12 +28,10 @@ V = ℝ^3
 typeof(V)
 V == CartesianSpace(3)
 supertype(CartesianSpace)
-supertype(EuclideanSpace)
-supertype(InnerProductSpace)
 supertype(ElementarySpace)
 ```
 i.e. `ℝ^n` can also be created without Unicode using the longer syntax `CartesianSpace(n)`.
-It is subtype of `EuclideanSpace{ℝ}`, a space with a standard (Euclidean) inner product
+It is subtype of `ElementarySpace{ℝ}`, with a standard (Euclidean) inner product
 over the real numbers. Furthermore,
 ```@repl tutorial
 W = ℝ^3 ⊗ ℝ^2 ⊗ ℝ^4

src/TensorKit.jl

@@ -17,7 +17,8 @@ export Fermion, FermionParity, FermionNumber, FermionSpin
 export FibonacciAnyon
 export IsingAnyon
 
-export VectorSpace, Field, ElementarySpace, InnerProductSpace, EuclideanSpace # abstract vector spaces
+export VectorSpace, Field, ElementarySpace # abstract vector spaces
+export InnerProductStyle, NoInnerProduct, HasInnerProduct, EuclideanProduct
 export ComplexSpace, CartesianSpace, GeneralSpace, GradedSpace # concrete spaces
 export ZNSpace, Z2Space, Z3Space, Z4Space, U1Space, CU1Space, SU2Space
 export Vect, Rep # space constructors

src/spaces/cartesianspace.jl

@@ -1,11 +1,11 @@
 """
-    struct CartesianSpace <: EuclideanSpace{ℝ}
+    struct CartesianSpace <: ElementarySpace{ℝ}
 
 A real Euclidean space `ℝ^d`, which is therefore self-dual. `CartesianSpace` has no
 additonal structure and is completely characterised by its dimension `d`. This is the
 vector space that is implicitly assumed in most of matrix algebra.
 """
-struct CartesianSpace <: EuclideanSpace{ℝ}
+struct CartesianSpace <: ElementarySpace{ℝ}
     d::Int
 end
 CartesianSpace(d::Integer = 0; dual = false) = CartesianSpace(Int(d))
@@ -28,6 +28,10 @@ function CartesianSpace(dims::AbstractDict; kwargs...)
     end
 end
 
+InnerProductStyle(::Type{CartesianSpace}) = EuclideanProduct()
+
+isdual(V::CartesianSpace) = false
+
 # convenience constructor
 Base.getindex(::RealNumbers) = CartesianSpace
 Base.:^(::RealNumbers, d::Int) = CartesianSpace(d)

src/spaces/complexspace.jl

@@ -1,13 +1,13 @@
 """
-    struct ComplexSpace <: EuclideanSpace{ℂ}
+    struct ComplexSpace <: ElementarySpace{ℂ}
 
 A standard complex vector space ℂ^d with Euclidean inner product and no additional
 structure. It is completely characterised by its dimension and whether its the normal space
 or its dual (which is canonically isomorphic to the conjugate space).
 """
-struct ComplexSpace <: EuclideanSpace{ℂ}
-  d::Int
-  dual::Bool
+struct ComplexSpace <: ElementarySpace{ℂ}
+    d::Int
+    dual::Bool
 end
 ComplexSpace(d::Integer = 0; dual = false) = ComplexSpace(Int(d), dual)
 function ComplexSpace(dim::Pair; dual = false)
@@ -29,6 +29,8 @@ function ComplexSpace(dims::AbstractDict; kwargs...)
     end
 end
 
+InnerProductStyle(::Type{ComplexSpace}) = EuclideanProduct()
+
 # convenience constructor
 Base.getindex(::ComplexNumbers) = ComplexSpace
 Base.:^(::ComplexNumbers, d::Int) = ComplexSpace(d)
@@ -42,17 +44,23 @@ Base.axes(V::ComplexSpace) = Base.OneTo(dim(V))
 Base.conj(V::ComplexSpace) = ComplexSpace(dim(V), !isdual(V))
 
 Base.oneunit(::Type{ComplexSpace}) = ComplexSpace(1)
-⊕(V1::ComplexSpace, V2::ComplexSpace) = isdual(V1) == isdual(V2) ?
-    ComplexSpace(dim(V1)+dim(V2), isdual(V1)) :
-    throw(SpaceMismatch("Direct sum of a vector space and its dual does not exist"))
-fuse(V1::ComplexSpace, V2::ComplexSpace) = ComplexSpace(V1.d*V2.d)
+function ⊕(V1::ComplexSpace, V2::ComplexSpace)
+    return isdual(V1) == isdual(V2) ?
+           ComplexSpace(dim(V1) + dim(V2), isdual(V1)) :
+           throw(SpaceMismatch("Direct sum of a vector space and its dual does not exist"))
+end
+fuse(V1::ComplexSpace, V2::ComplexSpace) = ComplexSpace(V1.d * V2.d)
 flip(V::ComplexSpace) = dual(V)
 
-infimum(V1::ComplexSpace, V2::ComplexSpace) = isdual(V1) == isdual(V2) ?
-    ComplexSpace(min(dim(V1), dim(V2)), isdual(V1)) :
-    throw(SpaceMismatch("Infimum of space and dual space does not exist"))
-supremum(V1::ComplexSpace, V2::ComplexSpace) = isdual(V1) == isdual(V2) ?
-    ComplexSpace(max(dim(V1), dim(V2)), isdual(V1)) :
-    throw(SpaceMismatch("Supremum of space and dual space does not exist"))
+function infimum(V1::ComplexSpace, V2::ComplexSpace)
+    return isdual(V1) == isdual(V2) ?
+           ComplexSpace(min(dim(V1), dim(V2)), isdual(V1)) :
+           throw(SpaceMismatch("Infimum of space and dual space does not exist"))
+end
+function supremum(V1::ComplexSpace, V2::ComplexSpace)
+    return isdual(V1) == isdual(V2) ?
+           ComplexSpace(max(dim(V1), dim(V2)), isdual(V1)) :
+           throw(SpaceMismatch("Supremum of space and dual space does not exist"))
+end
 
 Base.show(io::IO, V::ComplexSpace) = print(io, isdual(V) ? "(ℂ^$(V.d))'" : "ℂ^$(V.d)")