Abstract
In this chapter the class of compact linear operators is introduced, and studied in detail for the case of a Hilbert space. It is shown that the compactness condition is equivalent to the property that the operator maps weakly convergent to strongly convergent sequences, and that a uniform limit of compact operators is also compact. Properties of the spectrum of a compact operator are proved (Riesz-Schauder theory), and additional theory is presented for the case of compact self-adjoint operators, including the important result that there exists an orthonormal basis of the Hilbert space consisting of eigenvectors. The singular value decomposition of a compact operator is also discussed.
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