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Does every Hilbert space have an identity?

  1. Oct 23, 2011 #1
    I am sure that my questions are stupid. If we have a Hilbert space H, what do we mean by the closed subspace of H. Also, Does every Hilbert space have an identity? :P.

    Could any one please clean to me these things .
    Thanks!
     
  2. jcsd
  3. Oct 23, 2011 #2
    A subspace of a vector space (and a Hilbert space is a vector space) is a nonempty set X such that

    • [itex]x,y\in X~\Rightarrow~x+y\in X[/itex]
    • [itex]x\in X,\alpha\in \mathbb{C}~\Rightarrow~\alpha x\in X[/itex]

    A Hilbert space also comes equipped with a norm:

    [tex]\|x\|=\sqrt{<x,x>}[/tex]

    and a set X is closed under the norm if for all convergent sequences in X it holds that the limit is in X.

    A closed subspace is something that is both a subspace and closed.

    What do you mean with identity?? It has a 0, which is the additive identity...
     
  4. Oct 23, 2011 #3
    Thank You.
    I mean 1.
     
  5. Oct 23, 2011 #4
    That doesn't really help me. What is 1 supposed to mean??
     
  6. Oct 23, 2011 #5
    1 is the multiplicative identity
     
  7. Oct 23, 2011 #6
    And since when does a Hilbert space have a multiplication??
     
  8. Oct 23, 2011 #7
    Oops, thank you, that is why the question is stupid.
     
  9. Oct 23, 2011 #8
    There are no stupid questions :tongue2:
     
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