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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1997 • Question 4

Question

Math 6420 • PDE Fall 1997 • Question 4
Let \(\{\phi_i\}\) be a (finite or infinite) orthonormal set in a Hilbert space. Prove that for every vector \(f\) in the space \[ \lVert f\rVert^2\le \sum_i |\langle f,\phi_i\rangle|^2. \] If inequality in above equation is replaced by strict equality, what can be said about the set \(\{\phi_i\}\)?

Raw LaTeX

Let \(\{\phi_i\}\) be a (finite or infinite) orthonormal set in a
Hilbert space. Prove that for every vector \(f\) in the space
\[
  \lVert f\rVert^2\le \sum_i |\langle f,\phi_i\rangle|^2.
\]
If inequality in above equation is replaced by strict equality, what can be said about the
set \(\{\phi_i\}\)?
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Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Orthonormal Bases and Total Sets Poincare Inequality