5.4.5 Tensor Product Space Vector Space Role

Tensor Product Space Vector Space Role is the treatment of V ⊗ W, once its carrier set, addition, and scalar action have established it as satisfying the vector space axioms, as an ordinary vector space in every subsequent argument — subject to linear maps, capable of being spanned by a basis, decomposable into subspaces — with no residual trace of its origin as a quotient distinguishing it from any other vector space of the same dimension.

What Being "An Ordinary Vector Space" Licenses

Linear Maps To and From V ⊗ W Behave as Expected

Once V ⊗ W is established as a vector space, linear maps into it and out of it are governed by the same general theory as linear maps between any two vector spaces: a linear map V ⊗ W → U is determined by its values on a basis, the kernel and image of such a map are subspaces, and the rank-nullity theorem applies without modification. None of this requires re-deriving anything specific to the tensor product's construction; it follows from V ⊗ W being a vector space, full stop.

Subspaces of V ⊗ W Are Ordinary Subspaces

A subset of V ⊗ W closed under the addition and scalar action already established is a subspace in the ordinary sense, and results about subspaces — sums, intersections, direct sum decompositions — apply to subspaces of V ⊗ W exactly as they would to subspaces of any other vector space, including subspaces that consist entirely of non-decomposable elements alongside decomposable ones.

Dimension Counting and Isomorphism Classification Apply Directly

Because dim(V ⊗ W) = dim(V) · dim(W) in the finite-dimensional case, V ⊗ W is isomorphic, as an abstract vector space, to F^{dim(V) · dim(W)}, exactly as any vector space of that dimension over F would be; nothing about being a tensor product distinguishes V ⊗ W from another vector space of the same dimension once only the vector space structure is considered, and the distinguishing information — which elements are decomposable, how the space relates to V and W individually — lies outside the vector space axioms themselves.

What the Vector Space Role Does Not Capture

Decomposability Is Extra Structure, Not Part of Being a Vector Space

The property of being decomposable, or the notion of tensor rank, is meaningful only by reference to the specific bilinear map ⊗: V × W → V ⊗ W that accompanies the space; an abstract vector space isomorphic to V ⊗ W, considered without this accompanying map, has no way to single out which of its elements would correspond to decomposable tensors. The vector space role, by design, sets this distinguishing structure aside.

The Accompanying Bilinear Map Is Additional Data

Whenever V ⊗ W's role as the target of the universal property is invoked — for instance, factoring a bilinear map through it — the argument uses not just the vector space structure but also the specific map ⊗, which is extra data beyond what "being a vector space" by itself supplies.

Why This Role Is Worth Stating Explicitly

Justifying the Free Use of General Vector Space Theory

Explicitly confirming that V ⊗ W plays the role of an ordinary vector space is what justifies applying general linear algebra — bases, dimension, rank-nullity, subspace decompositions — to tensor products without qualification throughout the rest of tensor algebra, rather than having to re-verify vector space behavior each time such a result is used.