If is compact and , then is injective exactly when it is surjective. Equivalently, every nonzero point of the spectrum of is an eigenvalue. Compactness turns an approximate eigenvector sequence into a convergent eigenvector sequence.
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The Fredholm alternative is a principle in functional analysis that relates to the solvability of certain linear operator equations, particularly in the context of compact operators on Banach spaces or Hilbert spaces. It is especially relevant when dealing with integral equations and partial differential equations.