Bochner integral of real and imaginary parts Let $f:X\to E$ be a function defined on a measure space $(X,\Sigma,\mu)$ and taking values in a Banach space $E$. Let $F$ be another Banach space and $T:E\to F$ be a continuous linear map. If $f$ is integrable, then $T f$ is integrable, and in this case we have $$T\int f=\int T f$$
Using this result one can show that $f:X\to \mathbb{K}^n$ is integrable if and only if each coordinate function $f_j$ is integrable, and in this case $\int f=(\int f_1,\dots,\int f_n )$.
Now the book am reading proves the following corollary:
Corollary. $f:X\to \mathbb{C}$ is integrable if and only if $Re (f)$ and $Im(f)$ are integrable, and in this case $\int f =\int Re (f) +i\int Im(f)$.
Proof. Identify $\mathbb{C}$ with $\mathbb{R}^2$ and apply the previous result.
At first glance it seems right, but isn't there a field problem? For the previous result to hold it seems that $\mathbb{K}^n$ with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ is to be regarded as vector space over $\mathbb{R}$ or $\mathbb{C}$ respectively. But in the corollary am considering a function taking values in the Banach space $\mathbb{R}^2$ over $\mathbb{R}^2$, not over $\mathbb{R}$.
Am I missing something?
 A: As pointed out in the comments the definition of Bochner integrability does not depend on the scalar field being $\mathbb{R}$ or $\mathbb{C}$:
Simple functions  $\sum_{n=1}^N x_n 1_{A_n}$ with $x_n \in E$ and $A_n\in \Sigma$ do not depend on the field (strictly speaking $1_{A_n}$ scalar valued but it does not matter whether we regard it as real or complex valued). The integral of such simple function is defined as   $\sum_{n=1}^N x_n \mu(A_n)$ which does not depend on the field being $\mathbb{R}$ or $\mathbb{C}$.
An arbitrary function $f:X\to E$ is integrable if and only if there exist an $L^1$ Cauchy sequence of integrable simple functions converging almost everywhere to $f$. In this case the integral is defined as the limit of the simple function integrals. Again nothing in this definition depends on the field.
Hence $f:X\to \mathbb{C}$ is integrable over $\mathbb{C}$ if and only if it is integrable over $\mathbb{R}$ and both integrals coincide in this case.
Now if we identify $\mathbb{C}$ with $\mathbb{R}^2$ and regard them as vector spaces over $\mathbb{R}$, then we can prove the corollary.
