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())
     }
 }