If the inner product is defined on V with dimension less than or equal to 3 as [tex]\left\langle f,g \right\rangle = \int_{0}^{1}f(x)g(x)dx[/tex], I'm trying to find D* such that [tex]\left\langle Df,g \right\rangle[/tex] = [tex]\left\langle f,D*g \right\rangle[/tex], and I thought I had a closed form of D*. If {1, x, (x^2)/2, (x^3)/3}. Call each element e_i, i = 0,...,3 form a basis for V, then [tex]\left\langle Df,g \right\rangle[/tex] = [tex]\left\langle f,D*g \right\rangle[/tex] holds given D* is defined as i times the (derivative of g(x)) if g(x) has degree greater than 0, but I can't think of a closed form so that it will work for all elements. Does anyone have any suggestions?(adsbygoogle = window.adsbygoogle || []).push({});

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# Adjoint of a Differential Operator

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