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Homework Help: Hilbert spaces

  1. May 10, 2007 #1
    1. The problem statement, all variables and given/known data
    i have {ej} is an orthonormal basis on a hilbert space
    S1 is the 1-dimensional space of e1 and
    S2 is the span of vectors ej + 2e(j+1)

    eventually i need to show that S1 + S2 is dense in H and also evaluate
    S2 for density and closedness

    2. Relevant equations

    i know the def. of closed, dense, spans, etc...

    3. The attempt at a solution

    well, i know that i need to show that S1+S2 is dense by showing that its closure = my orthornormal basis. i think S2 is closed but not dense, but can an undense set + a dense set be a dense set?
     
  2. jcsd
  3. May 10, 2007 #2
    e1 is in S1 and 0 is in S2, so e1 = e1 + 0 is in S1 + S2.
    Is e2 in S1 + S2?
     
  4. May 10, 2007 #3
    yes...e2 = e1 + e1 + 0?
     
  5. May 10, 2007 #4
    No, e1 + e1 is not e2. But (e1 + e2) - e1 is e2.
     
  6. May 10, 2007 #5
    then S2 is dense afterall, but not closed. ...at least from what i've worked out since.
     
  7. May 10, 2007 #6
    I haven't looked at this carefully, but my first impression is that you are correct. Do you need further help with this?
     
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