Basic operator theory
Basic Operator Theory provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential and integral equations, approximation theory, and numerical analysis. A textbook designed for senior undergraduate and graduate students, Basic Operator Theory begins with the geometry of Hilbert space and proceeds to the spectral theory for compact self-adjoint operators with a wide range of applications. Part of the volume is devoted to Banach spaces and operators acting on these spaces. Presented as a natural continuation of linear algebra, Basic Operator Theory provides a firm foundation in operator theory, an essential part of mathematical training for students of mathematics, engineering, and other technical sciences.
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BOUNDED LINEAR OPERATORS ON HILBERT
Continuity of a linear operator
Operators of finite rank
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A-invariant adjoint operator Ax,y Banach space basic system basis for H bounded linear operator called closed subspace codim compact operator compact self adjoint complex numbers complex valued function converges uniformly convex COROLLARY corresponding DEFINITION ds dt eigen eigenvalues eigenvectors and eigenvalues example exists a unique Find finite dimensional finite rank fixed point following result function f Given Hence Hilbert space implies inequality infinite inner product space integral operator invertible k=l K K kernel function Lebesgue integrable Lebesgue measurable Lemma Let H linearly independent matrix max Ax,x n n n normed linear space operator defined operator with kernel operators of finite orthogonal projection orthonormal basis orthonormal system positive integer PROOF Prove real numbers real valued function Schauder basis Section space H spectral theorem subset Suppose system of eigenvalues system of eigenvectors Theorem l.l tion unique solution vector space