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I have encountered a problem on which I am sure there is some background, which unfortunately I don't know anything about (so that I don't even know where to start).

Let $(M, d_M)$, $(N, d_N)$ be compact metric spaces. If needed, we can assume that they are compact Riemannian manifolds. Let $Lip (M)$ (resp. $Lip (N)$) be the Banach space of complex-valued Lipschitz functions on $M$ (resp. $N$), endowed with the norm:

$$\|g\|_{Lip (M)} := \|g\|_\infty + \sup_{\substack{(x,y) \in M^2 \\ x \neq y}} \frac{|g(x)-g(y)|}{d_M (x,y)}.$$

I can endow $M \times N$ with e.g. the metric $d_M+d_N$.

First question (qualitative): can we meaningfully write $Lip (M) \otimes Lip (N) \simeq Lip (M \times N)$? I am not used to working with tensor products, much less topological tensor products. That may be trivial, but I am stumped.

Second question (quantitative): Let $f \in Lip (M \times N)$ be generic. Let $\varepsilon > 0$. How well can we approximate $f$ by products of Lipschitz functions on $M$ and $N$? More precisely, if $C(\varepsilon)$ is the least $C \geq 0$ such that there exist $n \geq 0$ and $(g_1, \cdots, g_n) \in Lip (M)^n$, $(h_1, \cdots, h_n) \in Lip (N)^n$ with:

$$\|f-\sum_{k=1}^n g_k \otimes h_k \|_\infty \leq \varepsilon,$$

$$\sum_{k=1}^n \|g_k\|_{Lip(M)} \|h_k\|_{Lip(N)} \leq C,$$

then what is the behaviour of $C(\varepsilon)$ as $\varepsilon$ goes to $0$? The use of the supremum norm in the first equation may seem weird, but that is what occurs in the problem I am looking at. The best case would be if $C(\varepsilon)$ were bounded, but I don't know of the approach the subject.

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  • $\begingroup$ Grothendieck showed that there are at most $ 14 $ ways of equipping an algebraic tensor product of Banach spaces with a natural cross-norm. However, it may turn out that none of these ways yields a Banach-space isomorphism. $\endgroup$ – Transcendental Apr 25 '15 at 20:09

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