5.10.4 Tensor Product Infinite Dimension Case

Tensor Product Infinite Dimension Case is the extension of tensor product theory to vector spaces that do not have finite dimension, covering both the purely algebraic setting, where a Hamel basis of possibly infinite cardinality still produces a well-defined tensor product, and the analytic setting of Hilbert and Banach spaces, where a topological completion step is required and several genuinely inequivalent tensor product constructions can arise.

The Algebraic Tensor Product of Infinite-Dimensional Spaces

The purely algebraic construction of the tensor product — as a quotient of a free vector space by multilinearity relations, or via its universal property — makes no assumption of finite dimension and applies verbatim once infinite-dimensional spaces are allowed.

Hamel Bases and Cardinal Arithmetic

Every vector space, of any dimension, has a Hamel basis (a basis in the ordinary linear-algebraic sense, using only finite linear combinations), whose existence for infinite-dimensional spaces relies on the axiom of choice. If V and W have Hamel bases of infinite cardinalities κ and λ, the algebraic tensor product V ⊗ W has a Hamel basis of cardinality κ · λ (cardinal multiplication), directly generalizing the finite-dimensional dimension multiplication law to the infinite setting.

Elements Remain Finite Sums

Even though the ambient spaces are infinite-dimensional, every individual element of the algebraic tensor product remains, by construction, a finite sum of decomposable tensors; the algebraic tensor product does not itself introduce any notion of convergent infinite sums, which is the central limitation motivating the topological completions described below.

Hilbert Space Tensor Products

When V and W are infinite-dimensional Hilbert spaces, the algebraic tensor product is typically too small to be useful on its own, since it excludes elements that should intuitively be considered part of a complete tensor product space, such as infinite orthogonal sums of decomposable terms with square-summable coefficients.

Completion with Respect to the Inner Product

The Hilbert space tensor product V ⊗̂ W is defined by first equipping the algebraic tensor product with the inner product ⟨v ⊗ w, v′ ⊗ w′⟩ = ⟨v, v′⟩⟨w, w′⟩, extended bilinearly, and then completing the resulting inner product space with respect to the induced norm, in the same way the real numbers complete the rationals.

Orthonormal Basis of the Completion

If {ei} and {fj} are orthonormal bases of V and W respectively, the tensors ei ⊗ fj form an orthonormal basis of the completed Hilbert space tensor product, and every element of V ⊗̂ W is representable as a possibly infinite, square-summable combination

extending the finite-sum representation available in the purely algebraic case to a genuinely infinite, convergent one.

Banach Space Tensor Products and Cross Norms

For Banach spaces, which need not carry an inner product, completing the algebraic tensor product requires choosing a norm compatible with the tensor structure, and unlike the Hilbert space case, more than one reasonable such choice exists.

The Projective and Injective Norms

The projective tensor norm defines ‖t‖ as the infimum, over all finite decompositions of t into decomposable terms, of the sum of the products of the individual factor norms, giving the largest reasonable cross norm. The injective tensor norm instead embeds the tensor product into a space of bilinear forms and takes the operator norm there, giving a generally smaller cross norm. These two constructions, and the completions they produce, can genuinely differ once V or W is infinite-dimensional, in contrast to the finite-dimensional case, where all reasonable norms on the tensor product are topologically equivalent.

Consequence for the Definition of "The" Tensor Product

Because distinct norms can produce distinct completions, the phrase "the Banach space tensor product" is ambiguous without specifying which cross norm is intended, a subtlety with no counterpart in finite dimensions or in the Hilbert space case, where the natural inner-product-induced norm is essentially forced by the requirement of compatibility with the inner product structure.

Contrast Summary

The finite-dimensional multiplicative dimension law extends cleanly, at the level of cardinal arithmetic, to the purely algebraic infinite-dimensional tensor product; extending further to a topologically complete tensor product of infinite-dimensional normed or inner-product spaces introduces genuinely new analytic content — completion, convergence, and, in the Banach space case, a choice of norm — that has no analogue in the elementary, purely algebraic theory of finite-dimensional tensor products.

Illustrative Diagram

The inner rectangle marks the algebraic tensor product of finite sums, sitting densely inside the larger completed space that also contains genuinely infinite, convergent combinations.