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Simple proof of Complex Inner Product Space

  1. Oct 10, 2015 #1

    RJLiberator

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    Gold Member

    1. The problem statement, all variables and given/known data
    Prove that <v|0>=0 for all |v> ∈ V.

    2. Relevant equations


    3. The attempt at a solution

    This is a general inner product space.

    I break it up into 2 cases.
    Case 1: If |v> = 0, the proof is trivial due to inner space axiom stating <0|0> = 0.

    Case 2: If |v> =/= 0 then:
    I use <v|0> = Σv_i * 0_i
    and from here it is clear to see that the sum adds up to 0 as every component is multiplied by the 0 vector.

    My question: Is this a safe definition of the complex inner product? Am I OK to use the summation definition in this general proof?
    Second Question: Is the proof correct? Any reason why the 0 vector would need to be proven further to sum the components to 0?

    Thanks
     
  2. jcsd
  3. Oct 10, 2015 #2

    RJLiberator

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    Gold Member

    I think I have a better idea.

    <v|0> = <v|0v> (by multiplication by 0.)
    = 0<v|v> by inner product axiom 2
    =0
    and done!

    This is the scalar 0, and not the vector 0 as stated in the question.
     
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