Typical examples of inner products that I've seen are inner products between functions (eg. $\langle f, g\rangle = \int_\mathbb{R} \bar{f}(x) g(x) w(x) dx$ for a weight function $w(x)$) or between vectors (eg. $\langle u, v\rangle = \sum_{i} \bar{u_i} v_i$). These can be generalized by $\langle u, v\rangle = \int_\mathbb{R} \bar{u(x)} v(x) d\mu(x)$ for a measure $\mu(x)$ – in the first case it's just $d\mu(x) = w(x) dx$ and in the latter it's the counting measure).
The Riesz Representation theorem says that any linear functional can be represented as an inner product with a fixed (first) argument, but it doesn't say that the inner product has to be a one dimensional integral. Are there spaces or inner products where the inner product is (or is better represented) as a double integral, such as $\langle u, v\rangle = \int_{\mathbb{R}^2} u(x, y) v(x, y) d\mu^2(x, y)$? Or alternatively, are there inner products that do not even look like any integral (of any dimension)?
Best Answer
Perhaps not what you're looking for, but every inner product can be expressed/understood as an one-dimensional integral of some sort. We can assume that our inner product space is in fact a Hilbert space since we can pass to the completion.
Note that every Hilbert space $H$ has an orthonormal basis $E$. Then every vector $x \in H$ can be written as $$x = \sum_{e\in E} \langle x,e\rangle e.$$ It follows that for $x,y \in H$ the inner product can be expressed as a sum $$\langle x,y\rangle = \sum_{e \in E} \langle x,e\rangle \overline{\langle y,e\rangle}$$ and the latter is simply the standard inner product of $|E|$-tuples $(\langle x,e\rangle)_{e \in E}$ and $(\langle y,e\rangle)_{e \in E}$ in the Hilbert space $L^2(E, \mu)$ where $\mu$ is the counting measure on the set $E$.
Therefore for all $x,y \in H$ we have $$\langle x,y\rangle = \sum_{e \in E} \langle x,e\rangle \overline{\langle y,e\rangle} = \int_E \langle x,e\rangle \overline{\langle y,e\rangle}\,d\mu(e).$$