2.16.4 Tensor Infinite Tensor Space Construction

Tensor Infinite Tensor Space Construction is the procedure by which a tensor space is assembled from an infinite-dimensional vector space, following the same universal-property definition of the tensor product used in finite dimension but requiring an explicit decision, absent in the finite case, about whether to stop at the purely algebraic construction or to complete it into a topological space capable of holding infinite superpositions. It is the infinite-dimensional counterpart to the finite tensor space construction, and the two share their starting definition exactly; they diverge only in what the resulting space is able to contain and in what extra data must be supplied to build it.

The Algebraic Starting Point Is Unchanged

Same Universal Property

The algebraic tensor product V ⊗ W of two vector spaces, finite- or infinite-dimensional, is defined by the identical universal property: it is the vector space, unique up to isomorphism, through which every bilinear map out of V × W factors uniquely as a linear map. This definition makes no reference to dimension, so the construction of V ⊗ W as formal finite sums of simple tensors v ⊗ w, modulo bilinearity relations, proceeds exactly as in the finite-dimensional case even when V and W are infinite-dimensional.

Iterating to Build T^p_q(V)

Just as in the finite case, the full type (p, q) tensor space is obtained by iterating this two-factor product p times over V and q times over the dual space being used.

where V′ denotes whichever dual (algebraic or continuous) is appropriate to the context, a choice that has no analogue in finite dimension since the two duals there coincide.

Where the Constructions Diverge

Elements Remain Finite Sums, Even Though the Space Is Infinite

Every element of the algebraically constructed T^p_q(V) is, by construction, a finite sum of simple tensors, exactly paralleling the finite-dimensional case. What is new is that T^p_q(V) is now itself infinite-dimensional, since its basis, inherited from a Hamel basis of V and its dual, is infinite. The algebraic construction is therefore "the same recipe applied to bigger ingredients," and it terminates in a well-defined but infinite-dimensional space without needing any new machinery.

When the Algebraic Space Is Not Enough

Applications in analysis frequently need objects that the algebraic tensor product cannot represent: an integral kernel, a general bounded or compact operator, or the limit of a sequence of simple tensors that converges in norm but not after finitely many terms. None of these belong to the algebraic V ⊗ W when V and W are infinite-dimensional normed spaces, since the algebraic construction admits only finite sums.

Topological Completion as an Additional Construction Step

Choosing a Cross Norm

To build a tensor space that does contain such limits, one equips the algebraic tensor product with a norm satisfying the cross-norm condition ‖v ⊗ w‖ = ‖v‖ · ‖w‖ on simple tensors, and then takes the completion of the normed space with respect to that norm. Distinct reasonable choices of cross norm, such as the projective norm or the injective norm, generally yield distinct, non-isomorphic completed spaces, unlike the finite-dimensional setting where the tensor product is unique regardless of any norm.

The Hilbert Space Case

When V and W are Hilbert spaces, there is a canonical choice: extend the inner products on V and W to a bilinear pairing on V ⊗ W by setting ⟨v_1 ⊗ w_1, v_2 ⊗ w_2⟩ = ⟨v_1, v_2⟩⟨w_1, w_2⟩, and complete with respect to the induced norm. The result, denoted V ⊗̂ W, is again a Hilbert space, and this is the construction underlying, for instance, the space of two-particle wavefunctions built from two single-particle Hilbert spaces.

Diagram of the Two-Stage Construction

Relation to the Finite Case

Completion Is Automatic (and Invisible) in Finite Dimension

Every finite-dimensional normed space is already complete, and the algebraic tensor product of two finite-dimensional spaces is automatically finite-dimensional and hence automatically complete in any norm placed on it, so the second stage of this construction is trivial and produces nothing new there. This is why the finite tensor space construction requires no discussion of completion at all: the step exists in principle but coincides with the algebraic construction itself. In the infinite-dimensional setting the two stages are genuinely distinct, and the choice of whether, and how, to complete is an essential part of specifying which tensor space is intended.