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I am trying to prove the spectral decomposition theorem for normal compact operators. Now, my book says the space H is the closure of the direct sum of $F_{t}$ where we the $F_{t}$ are eigenspaces and we sum over all eigenvalues $t$.

My question concerns what happens when there are 2 linearly independent eigenvectors associated to one eigenvalue. The above says we can write any $x$ in H as

$x=a_{1}e_{1}+a_{2}e_{2}+...$ where the $a_{i}$ are scalars and the $e_{i}$ are the eigenvectors,

My question concerns what happens when there are 2 linearly independent eigenvectors associated to one eigenvalue. The above says we can write any $x$ in H as

$x=a_{1}e_{1}+a_{2}e_{2}+...$ where the $a_{i}$ are scalars and the $e_{i}$ are the eigenvectors,

__one__from each eigenspace. So this tells me that 2 eigenvectors associated to the same eigenvalue cannot be used in the same linear combination. BUt I know that's not right.
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