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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2004 • Question 3

Question

Math 6210 • Real Analysis 2004 • Question 3

Let \(u_1,u_2,\cdots ,u_n,\cdots \) be an orthonormal set of vectors in a Hilbert space \(H\). Show that for every \(x\in H\) \[ \lim _{n\to \infty }(x,u_n)=0. \]

Raw LaTeX

Let \(u_1,u_2,\cdots,u_n,\cdots\) be an orthonormal set of vectors in
a Hilbert space \(H\). Show that for every \(x\in H\)
\[
\lim_{n\to\infty}(x,u_n)=0.
\]

Topic Tags

Hilbert Spaces and Orthogonality