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Linear Functional Question

  1. May 5, 2006 #1
    A linear functional is a function g:V to F where V is a vector space over a field F such that if u and v are elements of V and a is an element of F, then g(u+v) = g(u) + g(v) and
    g(au) = ag(u)
    Let G be the space of all linear functionals on V. Then if [tex]\oplus_{1}[/tex] and [tex]\otimes_{1}[/tex] are (repectively) addition and scalar multiplication in V, [tex]\oplus_{2}[/tex] and [tex]\otimes_{2}[/tex] are (repectively) addition and multiplication in F, and [tex]\oplus_{3}[/tex] and [tex]\otimes_{3}[/tex] are (repectively) addition and scalar multiplication in G, am I correct in saying that the correct way is to write
    g(u[tex]\oplus_{1}[/tex]v) = g(u) [tex]\oplus_{3}[/tex] g(v) and
    g(a[tex]\otimes_{1}[/tex]u) = a[tex]\otimes_{3}[/tex]g(u)?

    (God I hope the latex worked out..it's my first time)
     
  2. jcsd
  3. May 5, 2006 #2
    No, this is not correct. g(u) and g(v) are elements of F. Therefore you should write:
    g(u[tex]\oplus_{1}[/tex]v) = g(u) [tex]\oplus_{2}[/tex] g(v)
    and
    g(a[tex]\otimes_{1}[/tex]u) = a[tex]\otimes_{2}[/tex]g(u)

    Are the following what you are looking for (where g, h[tex]\in[/tex]G)?
    (g[tex]\oplus_{3}[/tex]h)(u) = g(u) [tex]\oplus_{2}[/tex] h(u)
    and
    (a[tex]\otimes_{3}[/tex]g)(u) = a[tex]\otimes_{2}[/tex]g(u)
     
    Last edited: May 5, 2006
  4. May 5, 2006 #3
    Right! Forgot about that.
     
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