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A question on linearity of functionals

  1. Jun 10, 2007 #1
    Suppose we have a bounded linear functional f defined on L1 (the sequence space of all absolutely summable sequences) and we take the natural (Schauder) basis for L1, that is, the set of sequences (E1,E2,....,En,....) that have 1 in the n th position and everywere else zero. Pick x in L1.
    Then x=A1*E1+A2*E2+.... , for some scalars An.
    Do we need to justify the fact that

    f(x)=A1*f(E1)+A2*f(E2)+... ?

    In other words, do we need to justify that linearity is still applied even on infinite sums?
     
  2. jcsd
  3. Jun 10, 2007 #2

    matt grime

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    There is nothing to justify at all.
     
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