##  [Spectral Theorem (Compact Self-Adjoint Operators)](/spectral-theorem-compact-self-adjoint-operators) 

  ##  [Spectral Theorem (Compact Self-Adjoint Operators)](https://natural.quantumdictionary.io/spectral-theorem-compact-self-adjoint-operators-0) 

  

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**Natural & Formal Sciences Dictionary**

 







 

 

 

 



 

 

 

 

Definition

For a compact self-adjoint operator T on a Hilbert space H, the spectral theorem states that H admits an orthonormal basis of eigenvectors of T, the corresponding eigenvalues are real, countable (with possible accumulation at 0) and each nonzero eigenvalue has finite multiplicity; T is representable as a convergent series T = Σ λ_k ⟨·,u_k⟩ u_k.