Is the Fourier inversion theorem valid for vector valued functions?

Question

I am currently reading Nakahara's book and he uses the equation \begin{equation} f(x)=\frac{1}{(2\pi)^n}\int\mathrm{d}k\ e^{ikx}\int\mathrm{d}y\ e^{-iky}f(y) \end{equation} on page 507, which looks like the usual Fourier inversion theorem - but I think that $f$ is a function from $\mathbf{R}^n$ to $\mathbf{C}^N\otimes V$, where $V$ is a finite-dim. vector space. I know that it is possible to define an integral for vector valued functions (e.g. the Bochner integral), but is the Fourier inversion theorem still valid?

Answer 1

Here the vector space $\mathbb C^N\otimes V$ is finite-dimensional, so integration is easy; no need for "Bochner" or other sophisticated vector integrals. Define $u=\int_A f(x)\;dx$ if and only if for all linear functionals $\alpha$ we have $\alpha(u)=\int_A \alpha(f(x))\;dx$. Then this inversion formula follows from the inversion formula with one-dimensional range.