A Hilbert Space Problem Book
Written for the active reader with some background in the topic, this book presents problems in Hilbert space theory, with definitions, corollaries and historical remarks, hints, proofs, answers and constructions.
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Vectors 1 Limits of quadratic forms
Schwarz inequality 3 Representation of linear functionals 4 Strict convexity
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adjoint algebraic analytic function answer approximate point spectrum assertion assumption bilateral cardinal number closure commutator compact operator complex numbers Conclusion conjugation consider continuous convergence Corollary corresponding countable cyclic vector defined dense diagonal operator dimension direct sum easy eigenvalue element equal example exists finite follows function q functional Hilbert space hence Hermitian operators Hilbert space hyponormal implies inequality infinite infinite-dimensional Hilbert space initial space inner product invertible operator linear transformation mapping matrix measurable function necessary and sufficient nilpotent non-zero norm normal operators numerical range operator on H orthogonal complement orthonormal basis partial isometry polar decomposition polynomial positive integer positive number Problem projection prove quasinilpotent result scalar sequence Solution space H span spec spectral radius spectral theorem strongly subset sufficient condition Suppose theory Toeplitz operators trivial uniform boundedness unilateral shift unit ball unit disc unit vector unitarily equivalent unitary operator weak topology weakly weighted shift