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I know we can write $\langle f(x), g(x) \rangle = \int f(x)g(x)dx$ and if we discretize the latter we can write $ = tr(g^T f)$.

But how could one write this in the general case where $f$ and $g$ depend on $n$ independent variables? $\langle f(x_1,x_2,...,x_n), g(x_1,x_2,...,x_n) \rangle $ = ?

Are there any good resources where I can learn more about this kind of mathematics?

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    Isnt't $tr(g^Tf)$ a complicated way of writing $gf^T$? – 2017-02-08
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    Your definition for one variable works in any number of variables (it's just a multiple integral then). Indeed it works for functions on any measure space (modulo details I'm not the one to check). – 2017-02-08
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    @Mariano, I think $tr$ is "trace", not "transpose". – 2017-02-08
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    @gerry, yes, that is precisely what I had understood ;-) – 2017-02-08
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    @Mariano, oh, never mind then. – 2017-02-08

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Measure theory has means to unite the "discretized" and "continuous versions" and also cover all those pernicious examples that involve both. It also extends to all numbers of dimensions:

If $A$ is a set provided with a (countably) additive set function $\mu$ then we can define an inner product on the set $L_\mu^2(A)$ of all $\mu$-square-integrable functions from $A \to \Bbb R$ by $$\langle f, g \rangle = \int fg \,\,d\mu$$

If $A$ is a connected subset of $\Bbb R$, the $\mu$ is Lesbegue measure and the integration is just ordinary integeration.

If $A$ is a finite set, for example, the integers from $1$ to $n$, then we can defined $\mu$ to be the cardinality: for $C \subseteq A$, $\mu(C) = |C|$, the number of elements in $C$. With this definition $$\int fg\,\,d\mu = \sum_{i \in A} f(i)g(i)$$

You can also combine the two: Let $A = \Bbb R \cup C$ where $C$ is finite, and define $U \subseteq A$ to be measurable if $U \cap \Bbb R$ is measurable. Define $\mu(U)$ to be sum of the Lesbegue measure of $U \cap \Bbb R$ and $|U \cap C|$. Then $$\int fg\,\,d\mu = \int_{U \cap \Bbb R} fg dx + \sum_{i \in U\cap C} f(i)g(i)$$

What about your question? That is just a matter of defining $\mu$ for multiple dimensions. If you mean the variables $x_i$ to be continuous variables, then the integral is just higher dimensional integration: For example, when $A = \Bbb R^2$, $$\langle f, g\rangle = \iint f(x,y)g(x,y)\,\, dx dy$$

If the $x_i$ are to be discrete, then the set A of all tuples $(x_1, x_2, ..., x_n)$ within the domain is still a finite set, so the definition I gave above still works: $$\langle f,g \rangle = \sum_{i \in A} f(i)g(i)$$ The only difference is that $i$ happens to be a tuple instead of an integer. In particular, if $A = \{1, 2, ..., m\}^n$ then the sum can also be expressed as $$\sum_{i \in A} f(i)g(i) = \sum_{x_1 = 1}^m \sum_{x_2 = 1}^m ... \sum_{x_n = 1}^m f(x_1, x_2, ..., x_n)g(x_1, x_2, ..., x_n)$$

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    I'm sorry you didn't get any good references like you asked. The ones I know were rare when I met them and are unlikely to accessible now, so I didn't offer. Any good abstract measure theory book would cover the subjects I mentioned. In general, mathematicians deal with higher dimensions by not specifying a dimension that they are working in (except when that dimension is an integral part of what they are studying). It usually only takes very little more development to work in $n$ dimensions than in $2$. Dimension is mostly only important when switching from 1 to 2 and from finite to infinite. – 2017-02-15
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    That's ok. I'm working my way through the proof laid out in this paper: https://www.researchgate.net/profile/Francois_Ducastelle/publication/223034883_Generalized_Cluster_Description_of_Multicomponent_Systems/links/0fcfd50cef0124fbcd000000.pdf The inner product used here used here is over functions of n-dimensional vectors, so your explanation helped me a good deal. I will see if I can find some references, thank you for pointing me in the right direction. – 2017-02-15