I'm trying to prove that certain space, which can be obtained as an interpolation space, is separable. The fact that is separable is well known but i want to simplify it via interpolation. I haven't been able to find any information (and i read a lot of interpolation references) about the separability and interpolation. My guess is that is preserved but i don't know if that's true. My question is, if $F$ denotes the real or complex interpolation methods, $(,)_{\theta,q}, [,]_\theta$, and $E_0,E_1$ are a compatible couple of separable spaces, then $F(E_0,E_1)$ is separable?
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The answer is no. Let us work with spaces on $[0,1]$. Then the weak Lebesgue space $L^{2,\infty}$ can be obtained using the real interpolation functor from the compatible couple $(L^1, L^3)$ (this essentially follows from [1], Theorem 4.13 on page 225, though technically, if you work with the abstract definition using $K$-functionals, you need some extra work). On the other hand, $L^{2,\infty}$ clearly does not have absolutely continuous norm, whence Theorem 5.5 on page 27 of [1] implies that $L^{2,\infty}$ is not separable.
[1] Interpolation of Operators, C. Bennett and R. Sharpley
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$\begingroup$ Thank you very much! And what about the complex method, which is the one im most interested in $\endgroup$ Commented Oct 3 at 15:31
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1$\begingroup$ For the complex method, any example I can think of is always separable. This a long shot, but there are these lecture notes on duality for the complex interpolation method (link below). There is nothing explicit about separability, but duality arguments often hide some separability in them, so it might be worth taking a look at the proofs. arxiv.org/pdf/0803.3558 $\endgroup$– kiliroyCommented Oct 3 at 16:52
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$\begingroup$ Okey, thanks!! I will give it a try maybe something appears! $\endgroup$ Commented Oct 3 at 18:40
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$\begingroup$ I wonder if the $q=\infty$ case is an outlier. From Kalton and Montgomery-Smith's chapter on "Interpolation of Banach Spaces" in the Handbook of the Geometry of Banach Spaces, vol 2, Proposition 5.1 asserts that $(X_0,X_1)_{\theta,q}$ for $q\in (1,\infty)$ and $\theta\in(0,1)$ is isomorphic to "a quotient of a subspace of $\ell_q(X_0\oplus X_1)$". If $X_0$ and $X_1$ are separable, then so is $\ell_q(X_0\oplus X_1)$. The quotient seems to (if I am not mistaken) be by $X_0\cap X_1$, which embeds as a closed subspace of $X_0 \oplus X_1$ following Aronszajn and Gagliardo. ... $\endgroup$ Commented Oct 3 at 19:28
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$\begingroup$ ... which would suggest that when $q \in (1,\infty)$ the real interpolation space is in fact separable when the endpoints $X_0$ and $X_1$ are. // Kalton and Montgomery-Smith also has a second part in that proposition regarding complex interpolation; but that discussion is a bit more sketchy (referring to "some alternative formulations of the complex method" with no citations), so I don't know whether the quotient taken preserved separability or not. (Someone should ask for a reference that gives the complete proof of that proposition.) $\endgroup$ Commented Oct 3 at 19:32