In the classic text of J.Bergh and J.Lofstrom, Interpolation Spaces, the $J$-method of real interpolation defines a functor in the following way: For $0<\theta<1$ and $1\leq q\leq \infty$ we define the functor $J_{\theta,q}$ which associates to every compatible couple $\overline{E}=(E_0,E_1)$ the space $(E_0,E_1)_{\theta,q;J}$ as the space of elements $x\in \sum(\overline{E})$ which can be represented by $$x=\int_0^\infty \frac{u(t)}{t}\,dt,$$ (with convergence in $\sum(\overline{E})$), where $u$ is a measurable function taking values on $\Delta(\overline{E})$ and satisfies $$\Phi_{\theta,q}\left(J(t,u(t))\right)<\infty.$$ We define $$||x||_{\theta,q;J}=\operatorname{inf}_{u}\Phi_{\theta,q}\left(J(t,u(t))\right),$$ where the infimum is taken over all the measurable functions $u$ taking values on $\Delta(\overline{E})$ and satisfying $\Phi_{\theta,q}\left(J(t,u(t))\right)<\infty$. Here $$J(t,x)=\operatorname{max}\{||x||_{E_0},t||x||_{E_1}\},$$ for $t>0$ and $x\in E_0\cap E_1$. In the proof of Theorem 3.2.2 which states that the $J$-method is an exact interpolation functor of exponent $\theta$ the author states that for every admisible operator $T$ we have $$Tx=T\left(\int_0^\infty u(t)/tdt\right)=\int_0^\infty Tu(t)/tdt,$$ without any commentary or observation, which seems to be very arbitrary. Researching i found that it could be related with Bochner integrals (something that the authors do not comment in any moment and its not trivial), which verify the interchanging property with bounded operators. The problem is that these integrals are defined on Banach spaces and in principle, the $J$-method is a functor on normed spaces, not only Banach ones. Is a mistake of the authors and the J-method is only definable on Banach spaces? Or there's something I'm missing.
Thanks in advance!
