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Prove dual space has the direct sum decomposition

  1. Apr 16, 2012 #1
    I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.

    Any direction would be greatly appreciated!

    Mike


    1. The problem statement, all variables and given/known data

    Let V be a vector space,

    Let W1, ..., Wk be subspaces of V, and,

    Let Vj = W1 + ... + Wj-1 + Wj+1 + ... + Wk.

    Suppose that V = W1 [itex]\oplus[/itex] ... [itex]\oplus[/itex] Wk.

    Prove that the dual space V* has the direct-sum decomposition V* = Vo1 [itex]\oplus[/itex] ... [itex]\oplus[/itex] Vok.

    2. Relevant equations

    See above.

    3. The attempt at a solution

    Again, I apologize for not having any attempted work, but I have no idea how to even begin tackling this proof.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 16, 2012 #2

    micromass

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    What is the [itex]{}^\circ[/itex] in [itex]V_j^\circ[/itex]?
     
  4. Apr 16, 2012 #3
    What is the " o " in Voj?

    Definition.
    If V is a vector space over the field F and S is a subset of V, the annihilator of S is the set So of linear functionals f on V such that f(α) = 0 for every α in S.
    .
    .
    .
     
  5. Apr 16, 2012 #4

    micromass

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    OK. You must prove

    [tex]V^*=V_1^\circ \oplus ... \oplus V_n^\circ[/tex]

    What does that mean?? What is the definition of a direct sum? What is it you need to check?
     
  6. May 8, 2012 #5
    Thank you!

    By the skin of my teeth, some help from you, and the grace of God, I received the best grade I could have expected in Linear Algebra.

    Thanks, again!

    Joe
     
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