Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1999 • Question 5

Question

Math 6420 • PDE Fall 1999 • Question 5

Prove the following statement. In an arbitrary real Hilbert space \(H\) an orthonormal system \((\phi _1,\phi _2,\ldots ,\phi _n,\ldots )\) is complete if and only if the equality \[ \lVert f\rVert ^2=\sum _{n=1}^{\infty }\langle f,\phi _n\rangle ^2 \] holds for any \(f\in H\).

Raw LaTeX

Prove the following statement. In an arbitrary real Hilbert
space \(H\) an orthonormal system \((\phi_1,\phi_2,\ldots,\phi_n,\ldots)\) is complete if and
only if the equality
\[
  \lVert f\rVert^2=\sum_{n=1}^{\infty}\langle f,\phi_n\rangle^2
\]
holds for any \(f\in H\).

Topic Tags

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