Algebraic Tensor Products and Completion

The Hilbert tensor product is a completion of finite sums of product vectors. The completion is essential: a normalizable bipartite state can require infinitely many product terms even though its norm is finite. The construction below establishes the inner product and clarifies which universal property survives completion.

Helpful background. Tensor Products of Hilbert Spaces supplies the physical composition rule. Hilbert Space as State Space supplies norm completion and orthonormal expansion.

Bilinear relations define the algebraic tensor product

Let $\mathcal H_A,\mathcal H_B$ be complex Hilbert spaces. Start with formal finite linear combinations of symbols $x\otimes y$ and impose bilinearity, including

$$(\alpha x+\beta x')\otimes y =\alpha x\otimes y+\beta x'\otimes y, \qquad x\otimes(\alpha y)=\alpha(x\otimes y).$$

The resulting vector space is $\mathcal H_A\odot\mathcal H_B$. Its elements are finite sums $\sum_jx_j\otimes y_j$, but a displayed sum is not unique.

The algebraic universal property says that every bilinear map $b:\mathcal H_A\times\mathcal H_B\to V$ factors through a unique linear map on $\mathcal H_A\odot\mathcal H_B$. No norm or continuity has yet been imposed. In particular, this statement does not guarantee extension to any chosen completed tensor product.

The symbol $\odot$ will distinguish the algebraic space from its Hilbert completion $\otimes$. A simple tensor is one product vector; a general algebraic tensor is a finite sum of such vectors.

The product inner product is positive definite

Using inner products linear in their second argument, prescribe

$$\left\langle \sum_i x_i\otimes y_i,\sum_j u_j\otimes v_j \right\rangle =\sum_{i,j}\langle x_i,u_j\rangle \langle y_i,v_j\rangle.$$

Sesquilinearity makes this compatible with the bilinear relations, so the value depends on the tensors, not their displayed expressions. Positivity can be checked inside finite-dimensional subspaces, avoiding any assumption about a completed product basis.

For a given finite sum $z$, choose orthonormal bases $e_1,\ldots,e_r$ and $f_1,\ldots,f_s$ for the spans of its first and second factors. Rewrite

$$z=\sum_{a=1}^r\sum_{b=1}^s c_{ab}e_a\otimes f_b.$$

The prescribed form gives

$$\|z\|^2=\sum_{a,b}|c_{ab}|^2.$$

This is nonnegative and vanishes only if every coefficient vanishes, hence only if $z=0$ in the algebraic tensor product. For Hilbert spaces with positive-definite inner products, there is therefore no remaining nonzero null subspace to quotient out.

In particular,

$$\|x\otimes y\|=\|x\|\,\|y\|.$$

This identity does not say that the norm of a sum is the sum of the product norms. Interference between product terms is encoded in the full double sum.

Completion adds infinite-rank vectors

Complete the algebraic space in this norm. The resulting Hilbert space is $\mathcal H_A\otimes\mathcal H_B$, with the algebraic tensors as a dense subspace.

For countable orthonormal bases $(e_a)$ and $(f_b)$, their simple products form an orthonormal basis of the completion. Every vector has an expansion

$$\Psi=\sum_{a,b}c_{ab}e_a\otimes f_b, \qquad \sum_{a,b}|c_{ab}|^2<\infty,$$

converging in Hilbert norm. The squared norm is the coefficient sum. Finite rectangular truncations converge because the omitted square sum tends to zero; this is not a pointwise convergence assertion in a wavefunction representation.

As a concrete example on $\ell^2\otimes\ell^2$,

$$\Psi=C\sum_{n=1}^\infty\frac1n e_n\otimes e_n, \qquad C=\left(\sum_{n=1}^\infty n^{-2}\right)^{-1/2},$$

is normalized. It is not an algebraic tensor. Any finite sum $\sum_{j=1}^r x_j\otimes y_j$ has first-factor contractions contained in the finite-dimensional span of the $x_j$. Contracting this $\Psi$ against the second-factor basis produces the infinitely many independent vectors $Ce_n/n$. Completion has added a vector of infinite Schmidt rank.

For nonseparable factors the product basis may be uncountable. Each individual vector still has at most countably many nonzero coefficients in that basis, since its square-summable support is countable.

A bounded bilinear map need not extend

The Hilbert tensor norm is not the norm representing all bounded bilinear maps. On $\ell^2\times\ell^2$, the complex bilinear form

$$b(x,y)=\sum_nx_ny_n$$

obeys $|b(x,y)|\leq\|x\|\|y\|$. Its algebraic linearization $L$ nevertheless fails to be bounded for the Hilbert tensor norm. Set

$$z_N=\frac1{\sqrt N}\sum_{n=1}^N e_n\otimes e_n.$$

Then $\|z_N\|=1$, but $L(z_N)=\sqrt N$. Thus $L$ has no continuous extension to the Hilbert completion. The universal norm for all bounded bilinear maps is the projective tensor norm, a different construction.

What the Hilbert construction does guarantee is uniqueness up to a canonical unitary: any dense realization of the algebraic product with the prescribed inner product completes to the same Hilbert space. Its compatibility with bounded tensor-product operators is developed in Hilbert Space Tensor Products.

Exercises

1. For the normalized infinite-rank vector above, compute the norm error after retaining its first $N$ terms.

Solution

Orthogonality gives squared error $C^2\sum_{n>N}n^{-2}$. The integral comparison bounds it by $C^2/N$, so the vector error is at most $C/\sqrt N$. The truncated vector is unnormalized; dividing by its norm changes the approximation slightly and should be stated separately.

2. Explain why the natural map $f\otimes g\mapsto[(x,y)\mapsto f(x)g(y)]$ extends to a unitary $L^2(X,\mu)\otimes L^2(Y,\nu)\simeq L^2(X\times Y,\mu\times\nu)$ for sigma-finite measure spaces.

Solution

Fubini–Tonelli gives the product inner product on simple tensors, hence an isometry on finite sums. Products of finite-measure set indicators span a dense subspace of the product $L^2$ space under the sigma-finiteness hypothesis. The isometry therefore has dense range and extends to an onto unitary after completion.

References

• Dereziński, Jan, and Christian Gérard, Mathematics of Quantization and Quantum Fields, Cambridge University Press, 2013, Chapter 3 — tensor spaces and their Hilbert completions. Publisher.
• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013 — Hilbert tensor products and composite systems. Publisher.
• Ryan, Raymond A., Introduction to Tensor Products of Banach Spaces, Springer Monographs in Mathematics, Springer, 2002 — projective tensor norms and bounded bilinear maps. Publisher.