TITLE: Do there exist inner product spaces for families of real valued functions other than weighted integrals?
QUESTION [8 upvotes]: In transform theory we join linear algebra with analysis by defining scalar products for real valued functions with weighted integrals of products, for example:
$$\langle f,g\rangle_w = \int_{-\infty}^{\infty} w(t)f(t)g(t)dt$$
And in multiple dimensions as multivariate integrals:
$$\langle f,g\rangle_w = \int\cdots\int_{-\infty}^{\infty} w(t_1,\cdots,t_k)f(t_1,\cdots,t_k)g(t_1,\cdots,t_k)dt_1 \cdots dt_k$$
To my question, does there exist other ways to define inner products for families of real valued functions?

REPLY [4 votes]: If you don't specify real valued functions further, there is also a really strange example: $l^2(\mathbb{R})$, the space of functions $f:\mathbb{R} \to \mathbb{R}$ for which $\sum_{x\in\mathbb{R}} |f(x)|^2 < \infty$ with the scalar product
$$\langle f,g\rangle_{l^2(\mathbb{R})} := \sum_{x\in\mathbb{R}} f(x)g(x)$$
is a well defined Hilbert-space. (Neither the sums nor the small $l$ are a typo and you have to think a bit about well-definedness)
In particular the space only consists of functions $f$ such that $\{x\in \mathbb{R}: f(x) \neq 0\}$ is countable, which means that $f=0$ almost everywhere. It is also not separable and as a result not isomorphic to $l^2(\mathbb{N})$ and thus to any of the other examples (which are all isomorphic to each other).