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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1995 • Question 3

Question

Math 6420 • PDE Fall 1995 • Question 3

Let \(S=\{\phi _1,\phi _2,\ldots ,\phi _n,\ldots \}\) be a countable orthonormal set in a Hilbert space \(\mathcal {H}\). Prove that \(S\) is a complete orthonormal set if and only if it is a maximal orthonormal set; i.e., there exists no unit vector \(\psi \) such that \(\{\psi ,\phi _1,\phi _2,\ldots ,\phi _n,\ldots \}\) is an orthonormal set.

Raw LaTeX

Let \(S=\{\phi_1,\phi_2,\ldots,\phi_n,\ldots\}\) be a countable
orthonormal set in a Hilbert space \(\mathcal{H}\). Prove that \(S\) is a complete orthonormal
set if and only if it is a maximal orthonormal set; i.e., there exists no unit vector \(\psi\)
such that \(\{\psi,\phi_1,\phi_2,\ldots,\phi_n,\ldots\}\) is an orthonormal set.

Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Orthonormal Bases and Total Sets