Inner product on Hilbert Spaces

Question

It's an open question. How could you define an inner product for a product of noncontable Hilbert spaces?

Answer 1

When you have a family of Hilbert spaces $\mathcal H_\alpha$, $\alpha\in A$, a natural thing to do is to

1. Introduce a measure $\mu$ on the set $A$.
2. Consider the space of functions $f$ from $A$ into the formal disjoint union $\bigsqcup_\alpha \mathcal H_\alpha$. Require $f(\alpha)\in\mathcal H_\alpha$ for all $\alpha$. Also require $\int_A \|f(\alpha)\|_{\mathcal H_\alpha}^2\,d\mu(\alpha)<\infty$.
3. Define the inner product as $\langle f, g\rangle = \int_A \langle f(\alpha), g(\alpha)\rangle \,d\mu(\alpha)$.
4. Worry about measurability in the above integrals.

5. Decide that two cases are enough for practical purposes:

   5a. $\mu$ is a counting measure on $A$
   5b. All $\mathcal H_\alpha$ are the same space

6. Stop worrying about measurability and go on enjoying your life.