Thread: spectral theorem

Suppose V is a finite-dimensional inner product space over C and T:V->V is a self-adjoint linear transformation. Then V has an orthonormal basis consisting of eigenvectors of T.
What would happens if V is infinite-dimensional?
Does the proof of finite-dimensional still work for infinite-dimensional? Why?

Originally Posted by guin

Suppose V is a finite-dimensional inner product space over C and T:V->V is a self-adjoint linear transformation. Then V has an orthonormal basis consisting of eigenvectors of T.
What would happens if V is infinite-dimensional?
Does the proof of finite-dimensional still work for infinite-dimensional? Why?

It almost works, but you also need T to be compact. See wikipedia, and the book `functional analysis' by Rudin should cover this (although I don't have a copy to hand).

Curiously, this question is actually a question about functional analysis; you've left the field of algebra with one small step! I would suggest asking any further questions in the analysis forum...