# Linear Differential Operator/Inner Product

## Main Question or Discussion Point

Can anyone help me?

Q. An inner product is defined on the vector space V of all (real) polynomials (of arbitrary
degree) by

<f,g>=Int(x*e^(-x)*f(x)*g(x),x=o..infinity

Show that the linear differential operator

L=-x*(d^2/dx^2)-(2-x)*(d/dx):V->V

is symmetric with respect to this inner product.

Don't necessarily need the exact answer, just need to know how I'm meant to go about it.

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HallsofIvy
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Can anyone help me?

Q. An inner product is defined on the vector space V of all (real) polynomials (of arbitrary
degree) by

<f,g>=Int(x*e^(-x)*f(x)*g(x),x=o..infinity

Show that the linear differential operator

L=-x*(d^2/dx^2)-(2-x)*(d/dx):V->V

is symmetric with respect to this inner product.

Don't necessarily need the exact answer, just need to know how I'm meant to go about it.
Let f and g be a polynomials. Show that <Lf, g>= <f, Lg>.

Let f and g be a polynomials. Show that <Lf, g>= <f, Lg>.
Ah right, thanks.

So all I need to do is replace the f(x) in the integral with Lf(x) (and the same with g(x) on the other side)? I'll give it a go anyway. Cheers.