Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2008 • Question 2

Question

Math 6420 • PDE Fall 2008 • Question 2

  1. Explain what is meant by a Cauchy sequence, Completeness of a space, and a Hilbert space (you can assume the definitions of norm, inner product, and linear space are known).
  2. Prove that the set of continuous functions on the interval \([a,b]\) is complete with respect to the uniform norm \[ \lVert f\rVert =\max _{x\in [a,b]} |f(x)|. \]

Raw LaTeX

\begin{enumerate}
\item[(a)] Explain what is meant by a Cauchy sequence, Completeness of a space, and a Hilbert
space (you can assume the definitions of norm, inner product, and linear space are known).

\item[(b)] Prove that the set of continuous functions on the interval \([a,b]\) is complete with
respect to the uniform norm
\[
  \lVert f\rVert=\max_{x\in[a,b]} |f(x)|.
\]
\end{enumerate}

Topic Tags

Hilbert Spaces and Orthogonality Linear Programming and Optimization Matrix Norms and Induced Norms Point, Continuous, and Residual Spectrum