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Prove: An integral operator with a symmetric kernel is self-adjoint.

Proof. A*f*_{1} is the function

and the inner product (A*f*_{1}, *f*_{2}), which is equal to the integral of the product of this function with
*f*_{2}, is given by the formula

In the same way

We are now going to do a little trick. We are going to rename the variables in 2). We are going to change the name of x to y and change the name of y to x. This makes no substantive change in the equation. Equation 2) then becomes

We now see that the right members of 1) and 3) are the same except for the kernel k(x, y). But we are assuming that the kernels are symmetric and thus k(x, y) = k(y, x). Thus the right members are equal and

which is the condition that the operator be self-adjoint.

References

Mathematics, Its Content, Methods and Meaning. Vol. 3, Chap. XIX

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