4

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the common definition of the $\ell_p$ space.

So, my question is: Is a Hilbert space (i) "an inner product space that is complete with respect to the norm induced by the inner product" or (ii) "a complete space with respect to a norm induced by some inner product"?

Notice that (i) is the common definition and (ii) is a definition for which the mentioned statement makes sense.

Thanks.

Community
  • 1
Pedro
  • 18,817
  • 7
  • 65
  • 127
  • 5
    A Hilbert space is a Banach space whose norm satisfies the parallelogram law? – Daniel Fischer Oct 01 '13 at 19:26
  • 6
    The two definitions are equivalent; if a norm is induced by some inner product, that inner product is unique. – Nate Eldredge Oct 01 '13 at 19:34
  • @NateEldredge the distinction he is drawing is set-theoretic. Often when you define a structure (groups, vector space, etc.) you define it to be a tuple $(G, +, \times, ... etc.)$ The way one obtains a Banach space from a Hilbert space isn't to say that the Banach space IS the hilbert space, but only that the Banach space is "canonically derived" from the Hilbert space. He wants to know why things were set up to make talking about this correctly awkward. At least that's what I think he means. – Jeff Oct 01 '13 at 19:37
  • This problem also comes up often when descriptive set theorists want to talk about metrizable spaces vs. metric spaces. I think by now I just proceed sort of on autopilot wrt distinctions like this, trusting myself to notice whenever it is an issue that doesn't resolve itself, so to speak. But I think this level of pedantry is something that is a good thing to go through at least at some point. Towards the OP, I suggest seeing if you believe that, if required, you could transcribe every statement you see into something that makes sense. For instance, sometimes people mean (i), sometimes no – Jeff Oct 01 '13 at 19:39
  • 4
    @Jeff Metrizable vs metric is somewhat different, since the topology of a metrizable space does not uniquely determine the metric. – Harald Hanche-Olsen Oct 01 '13 at 19:42

1 Answers1

8

The point is that there is a canonical one-to-one correspondence between Hilbert spaces (i) and Hilbert spaces (ii), where (i), (ii) refer to the two definitions in your question.

Given a Hilbert space (i), if you take the norm derived from the inner product and “forget” the original inner product then you have a Hilbert space (ii).

If you have a Hilbert space (ii), you can determine the unique inner product inducing the norm by using the polarization identity, and this gives you a Hilbert space (i).

The above two constructions are each other's inverses, therefore nobody bothers to distinguish between the two “kinds” of Hilbert spaces.

(This answer is just an expanded version of Nate Eldredge's comment above.)

Harald Hanche-Olsen
  • 31,960