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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 3

Question

Math 6420 • PDE Fall 2016 • Question 3

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 Linear Programming and Optimization