Inner product on a vector-valued function space A possible inner product on a function space $F=\{f:\mathbb{R}\to\mathbb{R}\}$
$$\int f_1(x)f_2(x)dx$$
Think of a space $F=\{f:\mathbb{R}^n\to\mathbb{R}\}$. How might I define an inner product on such a space and, does it even make sense to do so?
I motivate my thought process with an example. Think of a series of a function, like Fouriers. By defining an inner-product (lets say F is Hilbert) we may develope our function in a different base (sin and cos in Fouriers' case). I am exploring a similar idea, but first I need a good definition of an inner product.
Am I correct in assuming that as long as my definition of the inner product on F follows the rules, such operations make sense?
Could the inner product be defined as
$$\int_\Omega f(\vec{\textbf{x}})g(\vec{\textbf{x}})d\textbf{x}$$
What would be a rigorous way to go about it? Through some meassure or something else? Is there a standard approach to this?
Please forgive any lapses in terminology, I am an amateur exploring his interest.
Edit:
Perhaps I should broaden my question to a more general space
$$F=\{f:\mathbb{R}^n\to\mathbb{R}^m\}$$
I leave that decision to the expertise of the one who is kind enough to answer.
 A: Let's assume that $F=\{f:\mathbb{R}^n\rightarrow\mathbb{R}^m\}$. First, I think you mean to say vector-valued function instead of multi-valued. A multi-valued functions gives you multiple outputs for the same input. A vector-valued function gives you a single, vector output for a single input. Since for each $\vec{x}\in \mathbb{R}^n$, an element $v_1\in F$ should return only one element $v_1(\vec{x})\in \mathbb{R}^m$, although this element has $m$ components. 
Second, yes you are correct that if what you define satsifies the axioms (rules) of an inner product then you have an inner product. 
The problem with what you have written is that, a priori, there isn't a way to multiply $f$ and $g$ together, since they are vector-valued functions whose range is in $\mathbb{R}^m$. However, we can use the inner product on $\mathbb{R}^m$ (the ordinary dot product for $m$-dimensional vectors) to define multiplication between two vectors in $\mathbb{R}^m$ and thus define the following operation, $\eta(f,g)$ on $F$ as follows,
\begin{equation*}
\eta(f,g)=\int_{\Omega}f(\vec{x})\cdot g(\vec{x})dx.
\end{equation*}
Now you just need to check that $\eta$ satisfies the axioms of an inner product. To do this, you take a few arbitrary elements, $v_1, v_2, v_3$, of $F$ and show things like additivity, etc.
