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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Fall 2022 • Question 1

Question

Math 6710 • Applied Linear Operators and Spectral Methods Fall 2022 • Question 1

Let \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that

\[\sum _{n=1}^{\infty } \|u_n - v_n\|^2 < 1.\]

Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total orthonormal sequence. Hint: Assume for contradiction that \((v_n)\) is not total.

Raw LaTeX

Let \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that

\[\sum_{n=1}^{\infty} \|u_n - v_n\|^2 < 1.\]


Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total orthonormal
sequence.
Hint: Assume for contradiction that \((v_n)\) is not total.

Topic Tags

Hilbert Spaces and Orthogonality Orthonormal Bases and Total Sets Sequence Spaces