Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1994 • Question 1

Question

Math 6420 • PDE Fall 1994 • Question 1

Prove that an orthonormal sequence \(\{\phi _1,\phi _2,\ldots ,\phi _n\}\) in a Hilbert space \(\mathcal {H}\) is complete if and only if Parseval’s relation \[ \lVert f\rVert ^2=\sum _{n=1}^{\infty }|\langle f,\phi _n\rangle |^2 \] holds for all vectors \(f\in \mathcal {H}\).

Raw LaTeX

Prove that an orthonormal sequence
\(\{\phi_1,\phi_2,\ldots,\phi_n\}\) in a Hilbert space \(\mathcal{H}\) is complete if and only
if Parseval's relation
\[
  \lVert f\rVert^2=\sum_{n=1}^{\infty}|\langle f,\phi_n\rangle|^2
\]
holds for all vectors \(f\in\mathcal{H}\).

Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Orthonormal Bases and Total Sets Sequence Spaces