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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 3

Question

Math 6210 • Real Analysis Fall 2008 • Question 3

Let \(H\) be a Hilbert space with inner product written as \(\langle f,g\rangle \), and let \(f,g\in H\) be two non-zero elements. Show that \(f=zg\) for some \(z\in \mathbb C^*\) if and only if there exists no \(h\in H\) with \[ \langle f,h\rangle =1 \quad \text {and}\quad \langle g,h\rangle =0. \]

Raw LaTeX

Let \(H\) be a Hilbert space with inner product written as
\(\langle f,g\rangle\), and let \(f,g\in H\) be two non-zero elements. Show that
\(f=zg\) for some \(z\in\mathbb C^*\) if and only if there exists no \(h\in H\) with
\[
\langle f,h\rangle=1
\quad\text{and}\quad
\langle g,h\rangle=0.
\]

Topic Tags

Hilbert Spaces and Orthogonality Zeros of Holomorphic Functions and the Identity Theorem