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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2019 • Question 1

Question

Math 6420 • PDE Fall 2019 • Question 1

Let \((u_n)\) and \((v_n)\) be two orthonormal sequences in a Hilbert space \(H\). Assume that \[ \sum _{n=1}^{\infty }\lVert u_n-v_n\rVert <1. \] 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
\[
  \sum_{n=1}^{\infty}\lVert u_n-v_n\rVert<1.
\]
Show that if \((u_n)\) is a total orthonormal sequence, then \((v_n)\) is also a total
orthonormal sequence.

\textbf{Hint:} Assume for contradiction that \((v_n)\) is not total.

Topic Tags

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