# Interpolation of product spaces

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.

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.