Editing Self-adjoint operator (section)

=== Formulation for symmetric operators ===

The [[spectral theorem]] applies only to self-adjoint operators, and not in general to symmetric operators. Nevertheless, we can at this point give a simple example of a symmetric (specifically, an essentially self-adjoint) operator that has an orthonormal basis of eigenvectors. Consider the complex Hilbert space ''L''<sup>2</sup>[0,1] and the [[differential operator]]
: <math>A = -\frac{d^2}{dx^2}</math>
with <math>\mathrm{Dom}(A)</math> consisting of all complex-valued infinitely [[differentiable function|differentiable]] functions ''f'' on [0, 1] satisfying the boundary conditions 
: <math>f(0) = f(1) = 0.</math>

Then [[integration by parts]] of the inner product shows that ''A'' is symmetric.<ref group=nb>The reader is invited to perform integration by parts twice and verify that the given boundary conditions for <math>\operatorname{Dom}(A)</math> ensure that the boundary terms in the integration by parts vanish.</ref> The eigenfunctions of ''A'' are the sinusoids
: <math>f_n(x) =  \sin(n \pi x) \qquad  n= 1, 2, \ldots</math>
with the real eigenvalues ''n''<sup>2</sup>π<sup>2</sup>; the well-known orthogonality of the sine functions follows as a consequence of ''A'' being symmetric.

The operator ''A'' can be seen to have a [[compact operator on Hilbert space|compact]] inverse, meaning that the corresponding differential equation ''Af'' = ''g'' is solved by some integral (and therefore compact) operator ''G''. The compact symmetric operator ''G'' then has a countable family of eigenvectors which are complete in {{math|''L''<sup>2</sup>}}. The same can then be said for ''A''.