Suppose that $X\_{\theta}$ is an interpolation space between the Banach spaces $X\_0$ and $X\_1$. Let $\mathcal{B}$ be another Banach space.


**Is it true that $X\_{\theta}\times\mathcal{B}$ is an interpolation space between $X\_{0}\times\mathcal{B}$ and $X\_{1}\times\mathcal{B}$?**


Thank you for any suggestion.



Interpolation of product spaces
Yes, interpolation on product spaces works componentwise, so $$\Bigl(\prod\_{i=1}^n X\_i,\prod\_{i=1}^n Y\_i\Bigr) = \prod\_{i=1}^n (X\_i,Y\_i)$$ for any interpolation functor $(\cdot,\cdot)$ even with equal norms for a fixed choice of $\ell\_p$ norm on the product spaces. This follows from the restriction/corestriction theorem in Section 1.2.4 in the bible book of Triebel (or alternatively from interpolation for complemented subspaces in Section 1.17.1): One shows that the $j$-th component of the interpolation space of products is exactly $(X\_j,Y\_j)$.


For this, let $R\_j$ be the mapping which extracts the $j$-th component of an $n$-tuple, and let $E\_j$ insert the $j$-th component in an $n$-vector of zeroes. Then $E\_jR\_j$ is a linear continuous projection and $E\_jR\_j\prod\_{i=1}^n X\_i$ is a complemented subspace which is isometrically isomorphic to $X\_j$; analogously for $Y\_j$. Hence, by the mentioned theorem(s), 
$$E\_jR\_j\Bigl(\prod\_{i=1}^n X\_i,\prod\_{i=1}^n Y\_i\Bigr) =(X\_j,Y\_j)$$
and this again isometrically.




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*Triebel, Hans*, Interpolation theory. Function spaces. Differential operators, Berlin: Deutscher Verlag des Wissenschaften. 528 p. M 87.50 (1978). [ZBL0387.46033](https://zbmath.org/?q=an:0387.46033).


