1
$\begingroup$

While learning about the inner product space, I became curious why it is defined separately?

In my opinion, there seems to be no difference between defining the inner product space separately and defining the inner product operation at the same time as the vector space.

Is it because it is natural to define the dot product separately? Or is it because there is a vector space that cannot be an inner product space?

$\endgroup$
4
  • 4
    $\begingroup$ You can equip any vector space (over reals or complex numbers) with an inner product, but it might not be “natural”. There might be a more natural choice of norm/topology on the space (say, on $C[0, 1]$, the more natural choice would be the sup norm, while under any inner product norm the space would not be complete), or there might be no natural choice of norm/topology on the space at all. $\endgroup$
    – David Gao
    Commented Jul 3 at 6:16
  • 1
    $\begingroup$ (If your vector space is not over reals or the complex numbers, then the notion of inner product no longer makes much sense - you can still talk about bilinear forms, I suppose, but it wouldn’t as well-behaved as over reals or complex numbers. But vector spaces make perfect sense, and are useful in practice, over fields other than reals and complex numbers.) $\endgroup$
    – David Gao
    Commented Jul 3 at 6:19
  • 1
    $\begingroup$ Hi @DavidGao. What do you mean by "under any inner product norm the space would not be complete"? Isn't this space continuum-dimensional and therefore isomorphic as a vector space to eg $L^2(\Bbb R)$, and hence does admit a Hilbert space structure? $\endgroup$
    – Izaak van Dongen
    Commented Jul 3 at 17:12
  • 1
    $\begingroup$ @IzaakvanDongen Ah, yes, you’re right. I was not thinking clearly and subconsciously assuming the inner product norm in question is either stronger or weaker than the sup norm. I guess I should have said under any “natural” inner product norm the space would not be complete. $\endgroup$
    – David Gao
    Commented Jul 3 at 17:15

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .