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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6710 • Applied Linear Operators and Spectral Methods

Spring 2024 • Question 4

Question

Math 6710 • Applied Linear Operators and Spectral Methods Spring 2024 • Question 4

Let \(M\) be a subset of a Hilbert space \(H\). Assume \(M\) is such that for any \(v, w \in H\) for which the equality

\[\langle v, x\rangle = \langle w, x\rangle \]

holds for all \(x \in M\), we must have \(v = w\). Show that \(M^\perp = \{0\}\).

Raw LaTeX

Let \(M\) be a subset of a Hilbert space \(H\). Assume \(M\) is such that for any \(v, w \in H\) for
which the equality

\[\langle v, x\rangle = \langle w, x\rangle\]

holds for all \(x \in M\), we must have \(v = w\). Show that \(M^\perp = \{0\}\).

Topic Tags

Hilbert Spaces and Orthogonality