nLab tensor product of Banach spaces

Tensor products of Banach spaces


Functional analysis

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Tensor products of Banach spaces


There are various norms that may be placed on the tensor product of the underlying vector spaces of two Banach spaces; the result is not usually complete, but of course we may take its completion. One of these, the projective tensor product, makes Ban (the category of Banach spaces and short linear maps) into a closed symmetric monoidal category, but there are others that still put useful structures on BanBan. If we start with Hilbert spaces, then there is a choice of norm that will make the result into a Hilbert space; then Hilb also becomes a symmetric monoidal category.


Let VV and WW be Banach spaces, and let VWV \otimes W be their tensor product as vector spaces. To define a tensor product of VV and WW as Banach spaces, we will place a norm on VWV \otimes W, making a normed vector space; the only difference in the following definitions is which norm to use. Then we take the completion V^WV {\displaystyle\hat{\otimes}} W, which is a Banach space.

Definition (projective tensor product)

Every element of VWV \otimes W may be written (in many different ways) as a formal linear combination of formal tensor products of elements of VV and WW (suppressing the symbol \otimes):

iα iv iw i. \sum_i \alpha_i v_i w_i .

Let the projective cross norm x π{\|x\|_\pi} of an element xx of VWV \otimes W be

x πinf{ i|α i|v i Vw i W|x= iα iv iw i}. {\|x\|_\pi} \coloneqq \inf \{ \sum_i {|\alpha_i|} {\|v_i\|_V} {\|w_i\|_W} \;|\; x = \sum_i \alpha_i v_i w_i \} .

Then the projective tensor product V^ πWV {\displaystyle\hat{\otimes}_\pi} W of VV and WW is the completion of VWV \otimes W under the projective cross norm.

Definition (injective tensor product)

If λ\lambda and μ\mu are linear functionals on VV and WW (respectively), then λμ\lambda \otimes \mu is a linear functional on VWV \otimes W. Let the injective cross norm x ϵ{\|x\|_\epsilon} of an element xx of VWV \otimes W be

x ϵsup{|(λμ)x||λ V *,μ W *1}. {\|x\|_\epsilon} \coloneqq \sup \{ {|(\lambda \otimes \mu)x|} \;|\; {\|\lambda\|_{V^*}}, {\|\mu\|_{W^*}} \leq 1 \} .

Then the injective tensor product V^ ϵWV {\displaystyle\hat{\otimes}_\epsilon} W of VV and WW is the completion of VWV \otimes W under the injective cross norm.

Definition (tensor product of Hilbert spaces)

If VV and WW are Hilbert spaces, then their norms determine and are determined by their inner products, so let us discuss inner products. The elements of VWV \otimes W are generated by elements of the form vwv w, so set

v 1w 1,v 2w 2v 1,v 2w 1,w 2 \langle{v_1 w_1, v_2 w_2}\rangle \coloneqq \langle{v_1, v_2}\rangle \langle{w_1, w_2}\rangle

and extend by linearity. We write the norm of an element xx of the inner product space VWV \otimes W as x σ{\|x\|_\sigma}. Then the tensor product V^ σWV {\displaystyle\hat{\otimes}_\sigma} W of the Hilbert spaces VV and WW is the completion of VWV \otimes W under this norm (or inner product).

Cross norms

Besides the specific norms defined above, we can define axioms of a reasonable norm on VWV \otimes W.

Definition (cross norm)

A cross norm on VV and WW is any norm χ\chi on VWV \otimes W such that:

  • vw χ=v Vw W{\|v \otimes w\|_\chi} = {\|v\|_V} {\|w\|_W} for any elements vv and ww of VV and WW (respectively);
  • λμ χ *=λ V *μ W *{\|\lambda \otimes \mu\|_{\chi^*}} = {\|\lambda\|_{V^*}} {\|\mu\|_{W^*}} for any bounded linear functionals λ\lambda and μ\mu on VV and WW (respectively).
Definition (uniform cross norm)

A uniform cross norm is an operation that takes two Banach spaces and returns a norm on their algebraic tensor product, naturally in the two spaces. Equivalently, it's a functor χ:Ban×BanNVect\chi\colon Ban \times Ban \to NVect that makes the following diagram commute (or fills it with a natural isomorphism):

Ban×Ban χ NVect Vect×Vect Vect \array { Ban \times Ban & \overset{\chi}\rightarrow & NVect \\ \downarrow & & \downarrow \\ Vect \times Vect & \underset{\otimes}\rightarrow & Vect }

A uniform cross norm is obviously desirable from the nPOV, but does it meet the analysts' needs for a cross norm? Yes:


A uniform cross norm assigns a cross norm to any two Banach spaces.

The specific cross norms from the previous section qualify as much as possible:


The projective and injective cross norms are uniform cross norms (and hence are in fact cross norms). The norm on the algebraic tensor product of two Hilbert spaces is also a cross norm.

As far as I can tell, the Hilbert-space cross norm σ\sigma doesn't apply to arbitrary Banach spaces, so it doesn't define a uniform cross norm as defined above; however, it does define a functor on Hilb×HilbHilb \times Hilb, so it's as uniform as could be expected.

Looking only at the general theory of cross norms, the projective and injective cross norms appear naturally:

Proposition (Ryan 2002, Prop. 6.1)

Let VV and WW be Banach spaces and χ\chi be any norm on VWV \otimes W. Then χ\chi is a cross norm if and only if

x ϵx χx π, {\|x\|_\epsilon} \leq {\|x\|_\chi} \leq {\|x\|_\pi},

for every xVWx \in V \otimes W.

That is, we have a poset of cross norms, and the projective and injective cross norms are (respectively) the top and bottom of this poset.

We therefore obtain the following relationship between ϵ\epsilon, σ\sigma, and π\pi:


If VV and WW are Hilbert spaces and xx is an element of VWV \otimes W, then

x ϵx σx π. {\|x\|_\epsilon} \leq {\|x\|_\sigma} \leq {\|x\|_\pi} .

Of course, any cross norm χ\chi on VV and WW allows us to form the Banach space V^ χWV {\displaystyle\hat{\otimes}_\chi} W, which may reasonably be called a tensor product of VV and WW; that's why we care.

Schmidt decomposition

The Schmidt decomposition is a way of expressing a pure state in the tensor product of two Hilbert spaces in terms of states of the two components:

Theorem (Nielsen and Chuang Theorem 2.7)

Let AA and BB be finite-dimensional Hilbert spaces. Let |ψ|\psi\rangle be a pure state of ABA \otimes B. Then there exist orthonormal families? {|i A} i\{ |i_A \rangle \}_i in AA and {|i B} i\{ |i_B \rangle \}_i in BB, and non-negative real numbers λ i\lambda_i, such that

|ψ= iλ i|i A|i B |\psi\rangle = \sum_i \lambda_i |i_A \rangle \otimes |i_B\rangle

and iλ i 2=1\sum_i \lambda_i^2 = 1.

The numbers λ i\lambda_i are called the Schmidt co-efficients of |ψ|\psi\rangle, and the families {|i A}\{ |i_A\rangle \} and {|i B}\{ |i_B\rangle \} the Schmidt bases for AA and BB.


The Schmidt number of |ψ|\psi\rangle is the number of non-zero Schmidt coefficients of |ψ|\psi\rangle.


Although most often stated for finite dimensional Hilbert spaces, the Schmidt decomposition theorem stated as above works for AA and BB Hilbert spaces of arbitrary dimension (even uncountable). A proof follows through a straightforward application of the polar decomposition and spectral theorems to the positive trace-class operator given by the partial trace of the projection operator onto ψ\psi (see MO: Schmidt Decomposition on infinite-dimensional Hilbert spaces.)

Foundational issues

We need the Hahn–Banach theorem for ϵ\epsilon to be a cross norm; but σ\sigma and π\pi work regardless. Possibly some of the other propositions rely on some other form of the axiom of choice; I haven't seen their proofs.


See also:

Exposition in

Last revised on June 24, 2022 at 16:28:38. See the history of this page for a list of all contributions to it.