In chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, in the section subtitled Some Calculus, the author introduces the complex vector space $\mathcal{P}$ of smooth functions $\mathbb{R}^n\longrightarrow C^{m}$ which are $2\pi$-periodic in each argument. He then proceeds to define an $L_2$ inner product on this space by $$<\phi,\psi>=\int_{Q}\phi\cdot\psi,~ \forall \phi,\psi\in\mathcal{P} $$ where $\phi\cdot\psi$ denotes hermitian product of the functions; $$\phi\cdot\psi=\sum_{i=1}^m\phi_i\bar\psi_i. $$ and $Q$ denots the open cube in $\mathbb{R}^n$, i.e $Q={x\in\mathbb{R}^n; 0<x_j(x)<2\pi, 1<j<n}.$ To my knowledge such a Hermitian product is a function $\mathbb{R}^n\longrightarrow \mathbb{C}$. My question is how is this integral defined, how do we know it exists? I am only familiar with countour integrals for complex functions, and Lebesgue integrals for real functions. Does warner mean here the Lebesgue integral defined using complex measure?
Slightly later on he also defines the $\xi$th Fourier coefficient of $\phi\in\mathcal{P}$ for each $n$ tuple of integers $\xi=(\xi_1,\xi_2,\ldots,\xi_n)$ by $$\phi_{\xi}=\frac{1}{(2\pi)^n}\int_Q \phi(x)e^{-ix\cdot\xi}. $$ This is a complex vector function, how does the author mean this to be defined?
The integral of a complex valued function is just defined as in the real case: $$ \int_Q f(x)\;dx=\int_Q u(x)+iv(x)\;dx=\int_Q u(x)\;dx+i\int_Qv(x)\;dx $$ Here $u(x)$ and $v(x)$ are real valued functions - the real and imaginary part.
Now you know how to define the integral - it remains to show that it is finite. The closure of $Q$ is compact. Therefore, $\phi(x) \cdot \psi(x)$ is a continuous function, as $\phi$ and $\psi$ are continous. As continuous functions are bounded on compact subsets, we can also assume $\sup_{x \in \bar{Q}} |\phi(x)|\leq C $ as well as $\sup_{x \in \bar{Q}} |\psi(x)|\leq C $ for some $C \geq 0$. Therefore, the integral exists since $$ \int_Q \psi \cdot \phi \le (\int_Q |\phi|^2)^{1/2}(\int_Q |\psi|^2)^{1/2} \leq (\int_Q C^2)^{1/2}(\int_Q C^2)^{1/2}=|Q|C^2 $$ The last part: From this context, you can assume that $$ \phi_i(\xi)=\int_Q \phi_i(x)\exp{-ix \cdot \xi}\; dx $$ where we denote by $\phi(x)_i$ the $i$-th compenent of the vector $\phi(x)$.
One final note: If the Riemann and Lebesgue integral both exist, they coincide. And since your integrand is assumed to be continuous on a compact set, it doesnt really matter which integral you use in this case.