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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2016 • Question 2

Question

Math 6210 • Real Analysis Fall 2016 • Question 2

Let \(H\) be a Hilbert space, \(V\subset H\) a closed subspace, and \(F\subset H\) a finite-dimensional subspace, with \(F\cap V=\{0\}\). Show that there is a positive constant \(\lambda <1\) such that \[ |\langle u,v\rangle |\le \lambda \,\lVert u\rVert \,\lVert v\rVert \] for all \(u\in F\) and \(v\in V\).

Raw LaTeX

Let \(H\) be a Hilbert space, \(V\subset H\) a closed subspace, and
\(F\subset H\) a finite-dimensional subspace, with \(F\cap V=\{0\}\). Show that
there is a positive constant \(\lambda<1\) such that
\[
|\langle u,v\rangle|\le \lambda\,\lVert u\rVert\,\lVert v\rVert
\]
for all \(u\in F\) and \(v\in V\).

Topic Tags

Hilbert Spaces and Orthogonality