Asked by a physicist:

In physics we use the fact that, given a self adjoint operator $A$, every element in Hilbert space can be written as "generalized linear combination" of the eigenstates of $A$:

$$\psi(x)=\sum a_k \psi_k(x) +\int d\lambda c(\lambda)\psi_\lambda(x)$$ (1)

Where the integral is over the continuous spectrum.

If the dimension of $H$ is finite, the spectral theorem guarantees that this expression is valid for all A self adjoint with no continuous part.

In the case of infinite dimension spectral theorem says that every $A$ self adjoint is unitary equivalent to a multiplication operator, ossia, via basis change, we have

$$UTU^{-1}(\psi)(x)= f(x)\psi(x)$$

In which way is this related to the possibility to write every state in the form 1?