# HG changeset patch
# User Tuomo Valkonen <tuomov@iki.fi>
# Date 1756912108 18000
# Node ID fd9dba51afd3cb5386c86a951ca24e12e1cacfc2
# Parent  b4a47e8e80d13329c46d45c8649c5f80377d5a42
OwnedSpace -> Principal

diff -r b4a47e8e80d1 -r fd9dba51afd3 src/bisection_tree/btfn.rs
--- a/src/bisection_tree/btfn.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/bisection_tree/btfn.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -66,7 +66,7 @@
     G::SupportType: LocalAnalysis<F, BT::Agg, N>,
     BT: BTImpl<N, F>,
 {
-    type OwnedSpace = Self;
+    type Principal = Self;
     type Decomp = BasicDecomposition;
 }
 
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/convex.rs
--- a/src/convex.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/convex.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -57,7 +57,7 @@
 /// The conjugate type has to implement [`ConvexMapping`], but a `Conjugable` mapping need
 /// not be convex.
 pub trait Prox<Domain: Space>: Mapping<Domain> {
-    type Prox<'a>: Mapping<Domain, Codomain = Domain::OwnedSpace>
+    type Prox<'a>: Mapping<Domain, Codomain = Domain::Principal>
     where
         Self: 'a;
 
@@ -65,15 +65,15 @@
     fn prox_mapping(&self, τ: Self::Codomain) -> Self::Prox<'_>;
 
     /// Calculate the proximal mapping with weight τ
-    fn prox<I: Instance<Domain>>(&self, τ: Self::Codomain, z: I) -> Domain::OwnedSpace {
+    fn prox<I: Instance<Domain>>(&self, τ: Self::Codomain, z: I) -> Domain::Principal {
         self.prox_mapping(τ).apply(z)
     }
 
     /// Calculate the proximal mapping with weight τ in-place
-    fn prox_mut<'b>(&self, τ: Self::Codomain, y: &'b mut Domain::OwnedSpace)
+    fn prox_mut<'b>(&self, τ: Self::Codomain, y: &'b mut Domain::Principal)
     where
         Domain::Decomp: DecompositionMut<Domain>,
-        for<'a> &'a Domain::OwnedSpace: Instance<Domain>,
+        for<'a> &'a Domain::Principal: Instance<Domain>,
     {
         *y = self.prox(τ, &*y);
     }
@@ -98,7 +98,7 @@
 impl<F, E, Domain> Mapping<Domain> for NormConstraint<F, E>
 where
     Domain: Space,
-    Domain::OwnedSpace: Norm<E, F>,
+    Domain::Principal: Norm<E, F>,
     F: Float,
     E: NormExponent,
 {
@@ -127,7 +127,7 @@
     E: HasDualExponent,
     F: Float,
     Domain: HasDual<F>,
-    Domain::OwnedSpace: Norm<E, F>,
+    Domain::Principal: Norm<E, F>,
     <Domain as HasDual<F>>::DualSpace: Norm<E::DualExp, F>,
 {
     type Conjugate<'a>
@@ -146,7 +146,7 @@
     E: HasDualExponent,
     F: Float,
     Domain: HasDual<F>,
-    Domain::OwnedSpace: Norm<E, F>,
+    Domain::Principal: Norm<E, F>,
     <Domain as HasDual<F>>::DualSpace: Norm<E::DualExp, F>,
 {
     type Conjugate<'a>
@@ -165,10 +165,10 @@
 impl<Domain, E, F> Prox<Domain> for NormConstraint<F, E>
 where
     Domain: Space,
-    Domain::OwnedSpace: Norm<E, F>,
+    Domain::Principal: Norm<E, F>,
     E: NormExponent,
     F: Float,
-    NormProjection<F, E>: Mapping<Domain, Codomain = Domain::OwnedSpace>,
+    NormProjection<F, E>: Mapping<Domain, Codomain = Domain::Principal>,
 {
     type Prox<'a>
         = NormProjection<F, E>
@@ -206,11 +206,11 @@
 impl<F, E, Domain> Mapping<Domain> for NormProjection<F, E>
 where
     Domain: Space,
-    Domain::OwnedSpace: ClosedSpace + Projection<F, E>,
+    Domain::Principal: ClosedSpace + Projection<F, E>,
     F: Float,
     E: NormExponent,
 {
-    type Codomain = Domain::OwnedSpace;
+    type Codomain = Domain::Principal;
 
     fn apply<I: Instance<Domain>>(&self, d: I) -> Self::Codomain {
         d.own().proj_ball(self.radius, self.exponent)
@@ -253,7 +253,7 @@
 impl<Domain, Predual, F: Float> Preconjugable<Domain, Predual, F> for Zero<Domain, F>
 where
     Domain: Normed<F>,
-    Predual: HasDual<F, Owned = Predual>,
+    Predual: HasDual<F, PrincipalV = Predual>,
 {
     type Preconjugate<'a>
         = ZeroIndicator<Predual, F>
@@ -292,7 +292,7 @@
 where
     F: Float,
     Domain: Space,
-    Domain::OwnedSpace: Normed<F>,
+    Domain::Principal: Normed<F>,
 {
     type Codomain = F;
 
@@ -305,7 +305,7 @@
 impl<Domain, F: Float> ConvexMapping<Domain, F> for ZeroIndicator<Domain, F>
 where
     Domain: Space,
-    Domain::OwnedSpace: Normed<F>,
+    Domain::Principal: Normed<F>,
 {
     fn factor_of_strong_convexity(&self) -> F {
         F::INFINITY
@@ -315,7 +315,7 @@
 impl<Domain, F: Float> Conjugable<Domain, F> for ZeroIndicator<Domain, F>
 where
     Domain: HasDual<F>,
-    Domain::Owned: Normed<F>,
+    Domain::PrincipalV: Normed<F>,
 {
     type Conjugate<'a>
         = Zero<Domain::DualSpace, F>
@@ -331,7 +331,7 @@
 impl<Domain, Predual, F: Float> Preconjugable<Domain, Predual, F> for ZeroIndicator<Domain, F>
 where
     Domain: Space,
-    Domain::OwnedSpace: Normed<F>,
+    Domain::Principal: Normed<F>,
     Predual: HasDual<F>,
 {
     type Preconjugate<'a>
@@ -347,7 +347,7 @@
 
 impl<Domain, F> Prox<Domain> for ZeroIndicator<Domain, F>
 where
-    Domain: AXPY<Field = F, Owned = Domain> + Normed<F>,
+    Domain: AXPY<Field = F, PrincipalV = Domain> + Normed<F>,
     F: Float,
 {
     type Prox<'a>
@@ -430,7 +430,7 @@
     F: Float,
     X: Euclidean<F>,
 {
-    type Derivative = X::Owned;
+    type Derivative = X::PrincipalV;
 
     fn differential_impl<I: Instance<X>>(&self, x: I) -> Self::Derivative {
         x.into_owned()
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/direct_product.rs
--- a/src/direct_product.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/direct_product.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -311,7 +311,7 @@
     //     + for<'b> SubAssign<&'b Self>
     //     + Neg<Output = Self::OwnedEuclidean>,
 {
-    type OwnedEuclidean = Pair<A::OwnedEuclidean, B::OwnedEuclidean>;
+    type PrincipalE = Pair<A::PrincipalE, B::PrincipalE>;
 
     fn dot<I: Instance<Self>>(&self, other: I) -> F {
         other.eval_decompose(|Pair(u, v)| self.0.dot(u) + self.1.dot(v))
@@ -333,10 +333,10 @@
     F: Num,
 {
     type Field = F;
-    type Owned = Pair<A::Owned, B::Owned>;
+    type PrincipalV = Pair<A::PrincipalV, B::PrincipalV>;
 
     /// Return a similar zero as `self`.
-    fn similar_origin(&self) -> Self::Owned {
+    fn similar_origin(&self) -> Self::PrincipalV {
         Pair(self.0.similar_origin(), self.1.similar_origin())
     }
 
@@ -389,7 +389,7 @@
 pub struct PairDecomposition<D, Q>(D, Q);
 
 impl<A: Space, B: Space> Space for Pair<A, B> {
-    type OwnedSpace = Pair<A::OwnedSpace, B::OwnedSpace>;
+    type Principal = Pair<A::Principal, B::Principal>;
     type Decomp = PairDecomposition<A::Decomp, B::Decomp>;
 }
 
@@ -449,7 +449,7 @@
     }
 
     #[inline]
-    fn cow<'b>(self) -> MyCow<'b, Pair<A::OwnedSpace, B::OwnedSpace>>
+    fn cow<'b>(self) -> MyCow<'b, Pair<A::Principal, B::Principal>>
     where
         Self: 'b,
     {
@@ -457,7 +457,7 @@
     }
 
     #[inline]
-    fn own(self) -> Pair<A::OwnedSpace, B::OwnedSpace> {
+    fn own(self) -> Pair<A::Principal, B::Principal> {
         Pair(self.0.own(), self.1.own())
     }
 }
@@ -500,7 +500,7 @@
     }
 
     #[inline]
-    fn cow<'b>(self) -> MyCow<'b, Pair<A::OwnedSpace, B::OwnedSpace>>
+    fn cow<'b>(self) -> MyCow<'b, Pair<A::Principal, B::Principal>>
     where
         Self: 'b,
     {
@@ -508,7 +508,7 @@
     }
 
     #[inline]
-    fn own(self) -> Pair<A::OwnedSpace, B::OwnedSpace> {
+    fn own(self) -> Pair<A::Principal, B::Principal> {
         let Pair(ref u, ref v) = self;
         Pair(u.own(), v.own())
     }
@@ -601,7 +601,7 @@
 {
     type DualSpace = Pair<A::DualSpace, B::DualSpace>;
 
-    fn dual_origin(&self) -> <Self::DualSpace as VectorSpace>::Owned {
+    fn dual_origin(&self) -> <Self::DualSpace as VectorSpace>::PrincipalV {
         Pair(self.0.dual_origin(), self.1.dual_origin())
     }
 }
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/euclidean.rs
--- a/src/euclidean.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/euclidean.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -18,10 +18,11 @@
 /// as an inner product.
 // TODO: remove F parameter, use VectorSpace::Field
 pub trait Euclidean<F: Float = f64>:
-    VectorSpace<Field = F, Owned = Self::OwnedEuclidean>
-    + Reflexive<F, DualSpace = Self::OwnedEuclidean>
+    VectorSpace<Field = F, PrincipalV = Self::PrincipalE> + Reflexive<F, DualSpace = Self::PrincipalE>
 {
-    type OwnedEuclidean: ClosedEuclidean<F>;
+    /// Principal form of the space; always equal to [`Space::Principal`] and
+    /// [`VectorSpace::PrincipalV`], but with more traits guaranteed.
+    type PrincipalE: ClosedEuclidean<F>;
 
     // Inner product
     fn dot<I: Instance<Self>>(&self, other: I) -> F;
@@ -56,7 +57,7 @@
 
     /// Projection to the 2-ball.
     #[inline]
-    fn proj_ball2(self, ρ: F) -> Self::Owned {
+    fn proj_ball2(self, ρ: F) -> Self::PrincipalV {
         let r = self.norm2();
         if r > ρ {
             self * (ρ / r)
@@ -67,10 +68,10 @@
 }
 
 pub trait ClosedEuclidean<F: Float = f64>:
-    Instance<Self> + Euclidean<F, OwnedEuclidean = Self>
+    Instance<Self> + Euclidean<F, PrincipalE = Self>
 {
 }
-impl<F: Float, X: Instance<X> + Euclidean<F, OwnedEuclidean = Self>> ClosedEuclidean<F> for X {}
+impl<F: Float, X: Instance<X> + Euclidean<F, PrincipalE = Self>> ClosedEuclidean<F> for X {}
 
 // TODO: remove F parameter, use AXPY::Field
 pub trait EuclideanMut<F: Float = f64>: Euclidean<F> + AXPY<Field = F> {
@@ -89,5 +90,5 @@
 /// Trait for [`Euclidean`] spaces with dimensions known at compile time.
 pub trait StaticEuclidean<F: Float = f64>: Euclidean<F> {
     /// Returns the origin
-    fn origin() -> <Self as VectorSpace>::Owned;
+    fn origin() -> <Self as VectorSpace>::PrincipalV;
 }
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/instance.rs
--- a/src/instance.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/instance.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -113,8 +113,9 @@
 }
 
 /// Trait for abitrary mathematical spaces.
-pub trait Space: Ownable<OwnedVariant = Self::OwnedSpace> + Sized {
-    type OwnedSpace: ClosedSpace;
+pub trait Space: Ownable<OwnedVariant = Self::Principal> + Sized {
+    /// Principal, typically owned realisation of the space.
+    type Principal: ClosedSpace;
 
     /// Default decomposition for the space
     type Decomp: Decomposition<Self>;
@@ -131,8 +132,8 @@
 
 /// Helper trait for working with closed spaces, operations in which should
 /// return members of the same space
-pub trait ClosedSpace: Space<OwnedSpace = Self> + Owned + Instance<Self> {}
-impl<X: Space<OwnedSpace = Self> + Owned + Instance<Self>> ClosedSpace for X {}
+pub trait ClosedSpace: Space<Principal = Self> + Owned + Instance<Self> {}
+impl<X: Space<Principal = Self> + Owned + Instance<Self>> ClosedSpace for X {}
 
 #[macro_export]
 macro_rules! impl_basic_space {
@@ -141,7 +142,7 @@
     };
     ($type:ty where $($where:tt)*) => {
         impl<$($where)*> $crate::instance::Space for $type {
-            type OwnedSpace = Self;
+            type Principal = Self;
             type Decomp = $crate::instance::BasicDecomposition;
         }
 
@@ -215,7 +216,7 @@
 ///
 /// This is used, for example, by [`crate::mapping::Mapping::apply`].
 pub trait Instance<X, D = <X as Space>::Decomp>:
-    Sized + Ownable<OwnedVariant = X::OwnedSpace>
+    Sized + Ownable<OwnedVariant = X::Principal>
 where
     X: Space,
     D: Decomposition<X>,
@@ -235,7 +236,7 @@
         Self: 'b;
 
     /// Returns an owned instance of `X`, cloning or converting non-true instances when necessary.
-    fn own(self) -> X::OwnedSpace {
+    fn own(self) -> X::Principal {
         self.into_owned()
     }
 
@@ -244,7 +245,7 @@
     /// Returns an owned instance or reference to `X`, converting non-true instances when necessary.
     ///
     /// Default implementation uses [`Self::own`]. Consumes the input.
-    fn cow<'b>(self) -> MyCow<'b, X::OwnedSpace>
+    fn cow<'b>(self) -> MyCow<'b, X::Principal>
     where
         Self: 'b,
     {
@@ -255,7 +256,7 @@
     /// Evaluates `f` on a reference to self.
     ///
     /// Default implementation uses [`Self::cow`]. Consumes the input.
-    fn eval<'b, R>(self, f: impl FnOnce(&X::OwnedSpace) -> R) -> R
+    fn eval<'b, R>(self, f: impl FnOnce(&X::Principal) -> R) -> R
     where
         X: 'b,
         Self: 'b,
@@ -269,8 +270,8 @@
     /// Default implementation uses [`Self::cow`]. Consumes the input.
     fn either<'b, R>(
         self,
-        f: impl FnOnce(X::OwnedSpace) -> R,
-        g: impl FnOnce(&X::OwnedSpace) -> R,
+        f: impl FnOnce(X::Principal) -> R,
+        g: impl FnOnce(&X::Principal) -> R,
     ) -> R
     where
         Self: 'b,
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/linops.rs
--- a/src/linops.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/linops.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -28,35 +28,39 @@
 /// Vector spaces
 #[replace_float_literals(Self::Field::cast_from(literal))]
 pub trait VectorSpace:
-    Space<OwnedSpace = Self::Owned>
-    + Mul<Self::Field, Output = Self::Owned>
-    + Div<Self::Field, Output = Self::Owned>
-    + Add<Self, Output = Self::Owned>
-    + Add<Self::Owned, Output = Self::Owned>
-    + Sub<Self, Output = Self::Owned>
-    + Sub<Self::Owned, Output = Self::Owned>
+    Space<Principal = Self::PrincipalV>
+    + Mul<Self::Field, Output = Self::PrincipalV>
+    + Div<Self::Field, Output = Self::PrincipalV>
+    + Add<Self, Output = Self::PrincipalV>
+    + Add<Self::PrincipalV, Output = Self::PrincipalV>
+    + Sub<Self, Output = Self::PrincipalV>
+    + Sub<Self::PrincipalV, Output = Self::PrincipalV>
     + Neg
-    + for<'b> Add<&'b Self, Output = <Self as VectorSpace>::Owned>
-    + for<'b> Sub<&'b Self, Output = <Self as VectorSpace>::Owned>
+    + for<'b> Add<&'b Self, Output = <Self as VectorSpace>::PrincipalV>
+    + for<'b> Sub<&'b Self, Output = <Self as VectorSpace>::PrincipalV>
 {
+    /// Underlying scalar field
     type Field: Num;
-    type Owned: ClosedSpace
+
+    /// Principal form of the space; always equal to [`Space::Principal`], but with
+    /// more traits guaranteed.
+    type PrincipalV: ClosedSpace
         + AXPY<
             Self,
             Field = Self::Field,
-            Owned = Self::Owned,
-            OwnedVariant = Self::Owned,
-            OwnedSpace = Self::Owned,
+            PrincipalV = Self::PrincipalV,
+            OwnedVariant = Self::PrincipalV,
+            Principal = Self::PrincipalV,
         >;
 
     /// Return a similar zero as `self`.
-    fn similar_origin(&self) -> Self::Owned;
+    fn similar_origin(&self) -> Self::PrincipalV;
     // {
     //     self.make_origin_generator().make_origin()
     // }
 
     /// Return a similar zero as `x`.
-    fn similar_origin_inst<I: Instance<Self>>(x: I) -> Self::Owned {
+    fn similar_origin_inst<I: Instance<Self>>(x: I) -> Self::PrincipalV {
         x.eval(|xr| xr.similar_origin())
     }
 }
@@ -68,9 +72,9 @@
     + MulAssign<Self::Field>
     + DivAssign<Self::Field>
     + AddAssign<Self>
-    + AddAssign<Self::Owned>
+    + AddAssign<Self::PrincipalV>
     + SubAssign<Self>
-    + SubAssign<Self::Owned>
+    + SubAssign<Self::PrincipalV>
     + for<'b> AddAssign<&'b Self>
     + for<'b> SubAssign<&'b Self>
 where
@@ -93,8 +97,8 @@
     fn set_zero(&mut self);
 }
 
-pub trait ClosedVectorSpace: Instance<Self> + VectorSpace<Owned = Self> {}
-impl<X: Instance<X> + VectorSpace<Owned = Self>> ClosedVectorSpace for X {}
+pub trait ClosedVectorSpace: Instance<Self> + VectorSpace<PrincipalV = Self> {}
+impl<X: Instance<X> + VectorSpace<PrincipalV = Self>> ClosedVectorSpace for X {}
 
 /// Efficient in-place application for [`Linear`] operators.
 #[replace_float_literals(F::cast_from(literal))]
@@ -197,7 +201,7 @@
 }
 
 impl<X: Space> Mapping<X> for IdOp<X> {
-    type Codomain = X::OwnedSpace;
+    type Codomain = X::Principal;
 
     fn apply<I: Instance<X>>(&self, x: I) -> Self::Codomain {
         x.into_owned()
@@ -233,10 +237,10 @@
     }
 }
 
-impl<X: Clone + Space> Adjointable<X, X::OwnedSpace> for IdOp<X> {
-    type AdjointCodomain = X::OwnedSpace;
+impl<X: Clone + Space> Adjointable<X, X::Principal> for IdOp<X> {
+    type AdjointCodomain = X::Principal;
     type Adjoint<'a>
-        = IdOp<X::OwnedSpace>
+        = IdOp<X::Principal>
     where
         X: 'a;
 
@@ -245,10 +249,10 @@
     }
 }
 
-impl<X: Clone + Space> Preadjointable<X, X::OwnedSpace> for IdOp<X> {
-    type PreadjointCodomain = X::OwnedSpace;
+impl<X: Clone + Space> Preadjointable<X, X::Principal> for IdOp<X> {
+    type PreadjointCodomain = X::Principal;
     type Preadjoint<'a>
-        = IdOp<X::OwnedSpace>
+        = IdOp<X::Principal>
     where
         X: 'a;
 
@@ -262,9 +266,9 @@
 pub struct SimpleZeroOp;
 
 impl<X: VectorSpace> Mapping<X> for SimpleZeroOp {
-    type Codomain = X::Owned;
+    type Codomain = X::PrincipalV;
 
-    fn apply<I: Instance<X>>(&self, x: I) -> X::Owned {
+    fn apply<I: Instance<X>>(&self, x: I) -> X::PrincipalV {
         X::similar_origin_inst(x)
     }
 }
@@ -323,7 +327,7 @@
     where
         Self: 'b;
 
-    fn origin(&self) -> Y::Owned;
+    fn origin(&self) -> Y::PrincipalV;
     fn as_ref(&self) -> Self::Ref<'_>;
 }
 
@@ -334,7 +338,7 @@
         Self: 'b;
 
     #[inline]
-    fn origin(&self) -> Y::Owned {
+    fn origin(&self) -> Y::PrincipalV {
         return self.similar_origin();
     }
 
@@ -351,7 +355,7 @@
         Self: 'c;
 
     #[inline]
-    fn origin(&self) -> Y::Owned {
+    fn origin(&self) -> Y::PrincipalV {
         return self.similar_origin();
     }
 
@@ -392,8 +396,8 @@
     X: HasDual<F, DualSpace = Xprime>,
     Y: HasDual<F, DualSpace = Yprime>,
     F: Float,
-    Xprime: VectorSpace<Field = F, Owned = Xprime>,
-    Xprime::Owned: AXPY<Field = F>,
+    Xprime: VectorSpace<Field = F, PrincipalV = Xprime>,
+    Xprime::PrincipalV: AXPY<Field = F>,
     Yprime: Space + Instance<Yprime>,
 {
     pub fn new_dualisable(y_og: OY, xprime_og: OXprime) -> Self {
@@ -412,9 +416,9 @@
     F: Float,
     OY: OriginGenerator<Y>,
 {
-    type Codomain = Y::Owned;
+    type Codomain = Y::PrincipalV;
 
-    fn apply<I: Instance<X>>(&self, _x: I) -> Y::Owned {
+    fn apply<I: Instance<X>>(&self, _x: I) -> Y::PrincipalV {
         self.codomain_origin_generator.origin()
     }
 }
@@ -432,7 +436,7 @@
 impl<X, Y, OY, O, F> GEMV<F, X, Y> for ZeroOp<X, Y, OY, O, F>
 where
     X: Space,
-    Y: AXPY<Field = F, Owned = Y>,
+    Y: AXPY<Field = F, PrincipalV = Y>,
     F: Float,
     OY: OriginGenerator<Y>,
 {
@@ -450,7 +454,7 @@
 where
     X: Space + Instance<X> + Norm<E1, F>,
     Y: VectorSpace<Field = F>,
-    Y::Owned: Clone,
+    Y::PrincipalV: Clone,
     F: Float,
     E1: NormExponent,
     E2: NormExponent,
@@ -920,10 +924,10 @@
 where
     F: Float,
     Domain: Space,
-    Domain::OwnedSpace: Mul<F>,
-    <Domain::OwnedSpace as Mul<F>>::Output: ClosedSpace,
+    Domain::Principal: Mul<F>,
+    <Domain::Principal as Mul<F>>::Output: ClosedSpace,
 {
-    type Codomain = <Domain::OwnedSpace as Mul<F>>::Output;
+    type Codomain = <Domain::Principal as Mul<F>>::Output;
 
     /// Compute the value of `self` at `x`.
     fn apply<I: Instance<Domain>>(&self, x: I) -> Self::Codomain {
@@ -935,7 +939,7 @@
 where
     F: Float,
     Domain: Space,
-    Domain::OwnedSpace: Mul<F>,
-    <Domain::OwnedSpace as Mul<F>>::Output: ClosedSpace,
+    Domain::Principal: Mul<F>,
+    <Domain::Principal as Mul<F>>::Output: ClosedSpace,
 {
 }
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/loc.rs
--- a/src/loc.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/loc.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -456,7 +456,7 @@
 domination!(L2, L1);
 
 impl<F: Float, const N: usize> Euclidean<F> for Loc<N, F> {
-    type OwnedEuclidean = Self;
+    type PrincipalE = Self;
 
     /// This implementation is not stabilised as it's meant to be used for very small vectors.
     /// Use [`nalgebra`] for larger vectors.
@@ -739,7 +739,7 @@
 }
 
 impl<F: Num, const N: usize> Space for Loc<N, F> {
-    type OwnedSpace = Self;
+    type Principal = Self;
     type Decomp = BasicDecomposition;
 }
 
@@ -755,7 +755,7 @@
 
 impl<F: Float, const N: usize> VectorSpace for Loc<N, F> {
     type Field = F;
-    type Owned = Self;
+    type PrincipalV = Self;
 
     // #[inline]
     // fn make_origin_generator(&self) -> StaticEuclideanOriginGenerator {
@@ -763,12 +763,12 @@
     // }
 
     #[inline]
-    fn similar_origin(&self) -> Self::Owned {
+    fn similar_origin(&self) -> Self::PrincipalV {
         Self::ORIGIN
     }
 
     #[inline]
-    fn similar_origin_inst<I: Instance<Self>>(_: I) -> Self::Owned {
+    fn similar_origin_inst<I: Instance<Self>>(_: I) -> Self::PrincipalV {
         Self::ORIGIN
     }
 
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/nalgebra_support.rs
--- a/src/nalgebra_support.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/nalgebra_support.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -95,7 +95,7 @@
     DefaultAllocator: Allocator<N, M>,
     ShapeConstraint: StridesOk<E, N, M, SM> + StridesOk<E, N, M>,
 {
-    type OwnedSpace = OMatrix<E, N, M>;
+    type Principal = OMatrix<E, N, M>;
     type Decomp = MatrixDecomposition;
 }
 
@@ -270,10 +270,10 @@
     ShapeConstraint: StridesOk<E, M, N, S>,
 {
     type Field = E;
-    type Owned = OMatrix<E, M, N>;
+    type PrincipalV = OMatrix<E, M, N>;
 
     #[inline]
-    fn similar_origin(&self) -> Self::Owned {
+    fn similar_origin(&self) -> Self::PrincipalV {
         let (n, m) = self.shape_generic();
         OMatrix::zeros_generic(n, m)
     }
@@ -335,7 +335,7 @@
     ShapeConstraint: StridesOk<E, M, U1, SM>,
 {
     #[inline]
-    fn proj_ball(self, ρ: E, exp: Linfinity) -> <Self as Space>::OwnedSpace {
+    fn proj_ball(self, ρ: E, exp: Linfinity) -> <Self as Space>::Principal {
         let mut owned = self.into_owned();
         owned.proj_ball_mut(ρ, exp);
         owned
@@ -418,7 +418,7 @@
     DefaultAllocator: Allocator<M>,
     ShapeConstraint: StridesOk<E, M, U1, S>,
 {
-    type OwnedEuclidean = OVector<E, M>;
+    type PrincipalE = OVector<E, M>;
 
     #[inline]
     fn dot<I: Instance<Self>>(&self, other: I) -> E {
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/norms.rs
--- a/src/norms.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/norms.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -202,7 +202,7 @@
     F: Float,
     E: NormExponent,
     Domain: Space,
-    Domain::OwnedSpace: Norm<E, F>,
+    Domain::Principal: Norm<E, F>,
 {
     type Codomain = F;
 
@@ -232,18 +232,18 @@
 pub trait HasDual<F: Num = f64>: Normed<F> + VectorSpace<Field = F> {
     type DualSpace: Normed<F> + ClosedVectorSpace<Field = F>;
 
-    fn dual_origin(&self) -> <Self::DualSpace as VectorSpace>::Owned;
+    fn dual_origin(&self) -> <Self::DualSpace as VectorSpace>::PrincipalV;
 }
 
 /// Automatically implemented trait for reflexive spaces
 pub trait Reflexive<F: Num = f64>: HasDual<F>
 where
-    Self::DualSpace: HasDual<F, DualSpace = Self::OwnedSpace>,
+    Self::DualSpace: HasDual<F, DualSpace = Self::Principal>,
 {
 }
 
 impl<F: Num, X: HasDual<F>> Reflexive<F> for X where
-    X::DualSpace: HasDual<F, DualSpace = Self::OwnedSpace>
+    X::DualSpace: HasDual<F, DualSpace = Self::Principal>
 {
 }
 
diff -r b4a47e8e80d1 -r fd9dba51afd3 src/sets.rs
--- a/src/sets.rs	Wed Sep 03 09:52:30 2025 -0500
+++ b/src/sets.rs	Wed Sep 03 10:08:28 2025 -0500
@@ -51,8 +51,8 @@
         impl<U,Idx> Set<U> for $range<Idx>
         where
             U : Space<Decomp=BasicDecomposition>,
-            U::OwnedSpace : PartialOrd<Idx>,
-            Idx : PartialOrd + PartialOrd<U::OwnedSpace>,
+            U::Principal : PartialOrd<Idx>,
+            Idx : PartialOrd + PartialOrd<U::Principal>,
         {
             #[inline]
             fn contains<I : Instance<U>>(&self, item : I) -> bool {