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additive operator adjoint operator application Ax,x B-space Banach space Boundedness Cauchy sequence Chapter compact complete set complete space completely continuous operator complex number concept Consider continuous function converging sequence convex set coordinates corresponding countable dimension definition denoted distance function eigenvalues Euclidean space Example exists Function Space Functional Analysis fundamental sequence i=l i=l infinite dimensions inner product space inverse operator isomorphic kernel linear manifold linear normed space linear space linear vector space linearly independent linearly independent elements llxll mapping mathematical matrix metric space normed space open sphere operator A defined orthogonal orthonormal vectors positive operator problem proof properties rational numbers real numbers representation requirement satisfying scalar self-adjoint operator separable Hilbert space separable space set of orthonormal space H space of infinite specified subset symmetric operator theorem theory triangle inequality Trigonometric Polynomials vector space whence x e H zero element