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Prove set of sequences is a basis

  1. Sep 30, 2013 #1
    Let c_00 be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element x∈c_00, ∃N∈N such that xn=0,∀n≥n.

    Let {e_i}, i∈N be the set where e_i is the sequence in c_00 given by (e_i)_n =1 if n=i and (e_i)_n=0 if n≠i.

    Show that (e_i), i∈N is a basis for c_00.

    So I need to show it's linearly independent and that it spans c_00. I am not sure how to go about proving this makes it confusing is that it's an infinite set, so I can't use the usual method and take a finite number of vectors.

    I have an idea of how to prove linear independence, but not spanning.

    Any tips/hints?

    Thanks
     
  2. jcsd
  3. Sep 30, 2013 #2

    Zondrina

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    Hmm, I think contradiction would be good here.

    Suppose that ##\{e_i\}## is not a basis for ##C_∞##.

    What does that tell you about ##\{e_i\}##?
     
  4. Sep 30, 2013 #3

    tiny-tim

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    Hi SMA_01! :smile:
    I don't see the difficulty :confused: … for spanning, you need to prove that given any element, there's a finite number of basis elements that it is a linear combination of.
     
  5. Sep 30, 2013 #4
    What confused me was the fact that c_00 and {e_i} are infinite sets.
     
  6. Sep 30, 2013 #5

    tiny-tim

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    i] they're not sets :confused:

    ii] all you have to do is add a finite number of them …

    what difficulty would you have adding a finite number of decimal expansions? :smile:
     
  7. Sep 30, 2013 #6

    Zondrina

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    I would just like to make a side note that ##\{e_i\}## is a countably infinite set of sequences.

    ##C_∞## is an infinite dimensional subspace.
     
  8. Sep 30, 2013 #7
    Sorry, c_00 is a subspace, but {e_i} is a set.
    I understand now though how a finite number of the e_i's span any x in c_00, because x_n=0 for n≥N :smile:
     
  9. Sep 30, 2013 #8

    Zondrina

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    Yes, that's the idea.

    Since you know any sequence in ##C_∞## converges to zero (eventually the sequence terminates), it will always be possible to find a finite basis. You can scale this finite basis accordingly to represent any element in ##C_∞##.
     
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