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"If $V$ is a complex inner product space and $T\in \mathcal{L}(V)$. Then $V$ has an orthonormal basis Consisting of eigenvectors of T if and only if $T$ is normal".

  I know that the set of orthonormal vectors is called the "spectrum" and I guess that's where the name of the theorem. But what is the reason for naming it?

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Rather: the set of eigenvalues of a linear map is what is called spectrum.In the spectral theorem, you decompose the linear map in (very simple!) pieces, each piece coming from one element of the spectrum. –  Mariano Suárez-Alvarez Oct 22 '12 at 6:33
    
@Mariano, yes! My mistake, thanks. –  Hiperion Oct 22 '12 at 6:36
    
I'm teaching a Linear Algebra course at the moment and if things go according to schedule, I'll be discussing the Spectral Theorem on Halloween's day. :) –  Andrea Mori Oct 22 '12 at 10:36

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The name is provided by Hilbert in a paper published sometime in 1900-1910 investigating integral equations in infinite-dimensional spaces.

Since the theory is about eigenvalues of linear operators, and Heisenberg and other physicists related the spectral lines seen with prisms or gratings to eigenvalues of certain linear operators in quantum mechanics, it seems logical to explain the name as inspired by relevance of the theory in atomic physics. Not so; it is merely a fortunate coincidence.

Recommended reading: "Highlights in the History of Spectral Theory" by L. A. Steen, American Mathematical Monthly 80 (1973) pp350-381

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Thank you very much! –  Hiperion Oct 22 '12 at 7:54

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