Suppose we have a Hilbert space $H$. Is there any explicit expression for the inner product on $H^*$ without resorting to Riesz representation theorem?
I am NOT looking for one that uses the polarization identity - surely there must be a more "reasonable" expression?
Motivation: In some completely random context, I started with a normed vector space with norm coming from inner product, and was looking at its dual. Embarassingly I couldn't write down its inner product off the top of my head. I cannot come up with any "natural" candidate for the inner product, and while Riesz is nice, it certainly feels unncessarily convoluted, and that one should be able to write down a natural, explicit candidate.
A quick search yields this thread, but using polarization identity certainly doesn't help me "understand" what this inner product should really be and is thus quite unsatisfying. It would be nice if one can show me such an expression, or give me a reason why it shouldn't exist. Thanks!