Why do we want or need cross-norms on tensor products?

Question

If $(E, \|\cdot\|_1),(F, \|\cdot\|_2)$ are Banach spaces, their (algebraic) tensor product $E \otimes F$ is a vector space looking forward to be normed. A norm on a tensor product of vector spaces is called cross-norm if it behaves multliplicative on elementary tensors, i.e. for all $x \in E, y \in F$ we have $$ \|x \otimes y\| = \|x\|_1 \|y\|_2. \tag{1} $$

Prominent examples of cross-norms are the projective $\pi$ and injective $\varepsilon$ tensor product.

In my lecture notes I found that it is preferable to use cross-norms on tensor products. Somewhere else I found that one wants cross-norms since they are compatible with convergent sequences. This last staement does not convince me since I find it more natural to require only $$ \|x \otimes y\| \leq \|x\|_1 \|y\|_2 \tag{2}. $$ I think we all agree that a useful norm on a tensor product should fulfill (2): If $x_n \to x$ and $y_n \to y$, inequality $(2)$ implies that $x_n \otimes y_n \to x \otimes y$.

I have yet to find some occasion where I really need a norm fulfilling (1) and just (2) being not enough.

What do you think:

• Are cross-norms not only pretty but indispensable?. This should be taken care of by the addition I provided below: It seems natural to want the algebraic tensor product of the duals to be "included" in the dual of the algebraic tensor product, i.e. $E^* \otimes F^* \subseteq (E \otimes^\sim F)^*$ (if we identify a functional $\varphi \otimes \psi$ with its continuous extension to the completion $\cdot^\sim$ of $E \otimes F$ w.r.t. the cross-norm) although I am not capable of anticipating the consequences if this inclusion does not hold: What are the downsides of not having this inclusion? -- Is it reasonable not to use reasonable cross-norms? I think that $E^* \otimes F^*$ can be a nice point separating subspace of $(E \otimes^\sim F)^*$ but I am not sure about it.
• Can you provide examples of important norms for tensor products (of Banach spaces) that aren't (reasonable) cross-norms?

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Some more thoughts: Looking at Ryan's Introduction to Tensor Products p.127, reasonable norms on $E \otimes F$ are cross-norms in the sense of (1):

Let $(E,\|\cdot\|_1$ and $(F, \|\cdot\|_2)$ be Banach spaces. A norm $\|\cdot\|_\alpha$ on $E \otimes F$ is called reasonable cross-norm if

1. Inequality (2) holds.
2. For all $\varphi \in E^*$ and $\psi \in Y^*$, the linear functional $\varphi \otimes \psi$ on $X \otimes Y$ is bounded with operator norm $$ \|\varphi \otimes \psi\| \leq \|\varphi\| \|\psi\|. $$

Then, in proposition 6.1 it is proved that each reasonable cross-norm fulfills (1) automatically.

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In Category theory, there seems to be the concept of uniform cross norms, see nLab but from the definition there it is not clear why they call it uniform cross norm and not just uniform tensor product norm or something else.

Answer 1

I'd say this cross-norm property is important because it is precisely how the induced norm behaves in the case of the tensor product of Hilbert spaces. If $H_1 ≔ (U, +_U, ⋅_U, ⟨⋅∣⋅⟩_{H_1})$ and $H_2 ≔ (V, +_V, ⋅_V, ⟨⋅∣⋅⟩_{H_2})$ are Hilbert spaces over a field 𝕂, then $H₁⊗H₂$ is also a Hilbert space whose inner product is induced by

$$ ⟨u_1⊗v_1∣u_2⊗v_2⟩_{H_1⊗H_2} ≔ ⟨u_1∣u_2⟩_{H_1}⋅⟨v_1∣v_2⟩_{H_2} $$

This is the unique natural choice if you insist that $⟨u_i⊗v_j∣u_k⊗v_l⟩ = δ_{ik}δ_{jl}$ when $(u_i)$ and $(v_j)$ are orthogonal bases of $H_1$ and $H_2$ repectively.

Answer 2

Having a cross norm gives you separate continuity of the map $(\cdot)\otimes(\cdot): V\times V \to V\otimes V$. Fix a $w$ and take $\epsilon>0$. We want to see that there exists $\delta$ s.t. if $\|(v,w)- (v’,w)\|_{V\times V}<\delta$, then $\|v\otimes w - v’\otimes w \|_{V\otimes V}<\delta$

Assume $V$ is finite-dimensional and normed. Require $\|(v,w)- (v’,w)\|_{V\times V} = \|(v-v’,w)\|_{V\times V} < \delta$.

Let C be a constant of equivalence between the product space norm and the $L^1$-norm Then if $\delta:= \epsilon^{0.5 } C^{-1} $, we get:

$$\|v\otimes w - v’\otimes w \|_{V\otimes V} \\ = \|(v-v’)\otimes w \|_{V\otimes V} \\ \leq \|(v-v’) \|_{V} \|w \|_{V} \\ < \max \{\|(v-v’) \|_{V}, \|w \|_{V} \} ^2 \\ < C^2 \|(v-v’,w)\|_{V\times V} ^2 \\ < C^2\epsilon^{0.5 \cdot 2} C^{-2} \\ < \epsilon $$

This gives you half of the (in)equality for cross norms, the other half comes from the projection maps being continuous, and I will leave it for another time