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Completely Continuous Operators
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adjoint assume Banach space belongs boundary conditions bounded inverse Cauchy Chapter closed subspace complete orthogonal completely continuous operators complex numbers Consequently consider continuous functions converges uniformly corollary to Proposition define Definition eigenfunction of G eigenvalue eigenvectors equation Tx equivalent exists finite dimensional follows Fourier transform Fredholm given Green's function Hence Hermitian Hilbert problem Hilbert space Hilbert transform Hilbert-Schmidt infinite multiplicity inner product integral equation integral operator kernel K(x lemma Let F linear functional linear transformation logx Mercer's Theorem non-zero eigenvalues normed linear space null space obtain one-one operator with kernel orthonormal basis Parseval's identity Plancherel's Theorem prove Remark resolvent set satisfies self-adjoint sequence series converges spectral radius spectrum step functions subspace of H Suppose term by term tion unique solution vanishes Volterra whence x e H y e H zero