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Comment by ajtulloch | Hacker News Reader

For folks wondering about applications of this theorem: it is a key building block in the theory of reproducing kernel Hilbert spaces (RKHS), which in turn are the building block of kernel support vector machines (kernel SVMs), which are widely used in machine learning applications.

The "kernel trick" from kernel SVMs only works because of the existence and uniqueness result from the RRT on the underlying Hilbert space.

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There are several results called the Riesz representation theorem.

The article is about representing continuous linear functionals on a space of continuous functions as signed measures (or Riemann-Stieltjes integrals). This has lots of applications in ergodic theory or representation theory (e.g. disintegration of measures).

This result is essentially unrelated to the result characterizing continuous linear functionals on Hilbert spaces. It is also much more difficult to prove (the result on Hilbert spaces is rather simple).

If you can represent a continuous linear functional as the inner product with the Riesz representative then doesn't this also define a signed measure? It kind of seems like one of those theorems should imply the other to me, or is there some subtle aspect I'm missing?

It's the same thing really. It's just that Riesz first proved it for the special case and it was then generalized to Hilbert spaces. It's such a huge generalization that it causes a lot of confusion.

C([0,1]) is not a Hilbert space but a Banach space. Every Hilbert space is a Banach space, but not vice versa. The version of the theorem for Hilbert spaces is indeed a lot easier to prove than the one given in the article.

And that just proves my point about confusion :)

Thanks for the correction.

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