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Inner product space of continuous function

  1. Oct 28, 2011 #1
    1. The problem statement, all variables and given/known data
    C[a,b] is a vector space of continuos real valued functions. for f,g in C[a,b]
    <f,g>=∫f(x)g(x)dx, [a,b]

    Give a completely rigorous proof that if <f,f>=0, then f=0

    2. The attempt at a solution
    I tried to prove this by contrapositive, "f≠0 implies that <f,f>≠0

    When f(x)≠0, f(x)<0 or f(x)>0. However, [f(x)]2>0 in both cases. Therefore, there exist a<c<b and a real number d such that [f(c)]2=d > 0

    Since f is a continuous function, f2 is also a continuos function.
    By definition of continuity,
    For all ε>0, exist δ>0 such that
    for all x : |x-c|<δ implies that |f2(x)-f2(c)|<ε

    We choose ε<d/2 and get that
    -d/2<f2(x)-f2(c)<d/2
    0<d/2<f2(x)<3d/2
    So, for all x that is in the interval [c-δ,c+δ], f2(x)>0
    (can we conclude anything about the x not in this interval? like they'll be 0?)

    The following is the part where i'm not sure...
    To compute the integral ∫[itex]^{a}_{b}[/itex]f2(x)
    we divide [a,b] into n-subinterval
    a=x1<x2<...<xi-1<c-δ<c+δ<xi+1<...<xn=b
    Let x[itex]^{i}_{*}[/itex]= midpoint of each interval [x[itex]_{i}[/itex],x[itex]_{i+1}[/itex]] where we denote c-δ<x<c+δ
    ∫[itex]^{a}_{b}[/itex]f2(x)≈f[itex]^{2}[/itex](x[itex]_{1}[/itex])(x2-x1)+...+f[itex]^{2}[/itex](x)δ+...+f[itex]^{2}[/itex](x[itex]_{n-1}[/itex])(xn-xn-1)
    >f[itex]^{2}[/itex](x)δ>0 I'm really not sure about this step

    So since f≠0 implies <f,f>=∫[itex]^{a}_{b}[/itex]f2(x)≠0.<f,f>=0 implies f=0

    Thanks
     
  2. jcsd
  3. Oct 28, 2011 #2

    HallsofIvy

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    If f is not identically 0, then there exist some [itex]x_0[/itex] such that [itex]f(x_0)[/itex] is not 0. Since f is continuous, there exist some interval around [itex]x_0[/itex] such that f(x) is not 0 for any x in that interval. Also use the fact that [itex]f^2(x)\ge 0[/itex] for any real valued function, f, and all x.
     
  4. Oct 28, 2011 #3
    so did i do it correctly?
     
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