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Math Help - Functional non-multiplicative

  1. #1
    Junior Member
    Joined
    Oct 2009
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    70

    Functional non-multiplicative

    How can I show that the following functional is non-multiplicative:
    f(x)=\int_0^1{x(t)dt}
    (We are working in the space X=C(0,1))
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  2. #2
    Junior Member
    Joined
    Feb 2010
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    Lisbon
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    Let Id be the identity map on (0,1). Then

    f(Id\cdot Id)=\int_0^1t^2dt=\frac{1}{3}.

    However

    f(Id)\cdot f(Id)=\left ( \int_0^1 tdt\right )^2=\left( \frac{1}{2}\right)^2=\frac{1}{4},

    so f is not a multiplicative functional.
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