Hilbert, Schmidt and the Spectral Theorem
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The Spectral Theorem comes in many flavors, one might encounter it first in it’s finite dimensional case, for $\mathbb{C}^n$. If $T:\mathbb{C}^n \to \mathbb{C}^n$ is a self-adjoint linear map then there is a basis ${x_1, …, x_n}$ of $\mathbb{C}^n$ consisting of eigenvectors of $T$ with which we have: \(Tx= \sum_{i=1}^n \lambda_i \langle x, x_i\rangle x_i.\) As its name suggests, it falls to Spectral Theory to extend this result into infinite dimensional spaces. As it’s usually the case, this brings forth considerable difficulties. One way of easing these difficulties is to retain some notion of finitude, in this case compactness.