Spectral theorem of compact operators in Hilbert space I am reading the following theorem from my lecture notes (English translation of German text). But I don't understand exactly what is meant from this theorem and the proof.

Theorem.
Let $H$ be a Hilbert space and $T \in K(H)$ where $K(H)$ denotes the set of compact operators from $H$ to $H$. Let $T$ be normal or self adjoint depending on whether the underlying field is $\mathbb R$ or $\mathbb C$, then there exists an orthonormal set $\{e_i \mathrel| i\in I\}$ where $I$ is either $\mathbb N$ or $\{1,2,\dotsc,k\}$ and a sequence $(\lambda_i)_{i\in I} \in \mathbb K$ which converges to $0$  such that
$$\operatorname{span}{(x_i : i ∈ I})^{⊥} = \ker(T)$$
and also $\forall x \in H$ : $Tx= \sum_{i\in I} \lambda_i \langle x, e_i \rangle e_i$ with unconditional convergence.

I am sorry here because I don't understand the theorem completely which means the  translation is wrong, but can someone point out this theorem in some textbook or somewhere.
Thanks.

I guess the most important thing to realize here is that compact operators have only point spectrum formed of eigenvalues, which tend to 0 (and 0 is in the spectrum as well). Then we can write the operator as a countable sum over projections multiplied by eigenvalues (in such way that the eigenvalues tend to 0) opposed to the need of writing some operators as integrals over spectral measures.
