Math 6420 • PDE

Let \(X\) be a real Banach space. Assume \(f\in X^*\) has a closed nullspace
\(\mathcal{N}(f)\). The goal of this problem is to show that \(f\) must be a bounded linear
functional.
\begin{enumerate}
\item[(a)] Let \(x_0\in X\) be such that \(f(x_0)=1\). Explain why there is an \(\epsilon>0\)
for which the ball
\[
  B(x_0,\epsilon)\equiv\{x\in X\mid \lVert x-x_0\rVert<\epsilon\}
\]
satisfies \(B(x_0,\epsilon)\subset X-\mathcal{N}(f)\).

\item[(b)] Prove that \(f(x)>0\) for all \(x\in B(x_0,\epsilon)\), where \(\epsilon\) is as in
part (a). To prove this, you may assume for contradiction that there is some
\(y\in B(x_0,\epsilon)\) with \(f(y)<0\).

\item[(c)] Any \(x\in B(x_0,\epsilon)\) can be written as \(x=x_0+\epsilon u\) for some \(u\)
with \(\lVert u\rVert<1\). Use the result of part (b) to show that
\(\lvert f(u)\rvert<1/\epsilon\).

\item[(d)] To conclude, give an upper bound for \(\lVert f\rVert\).
\end{enumerate}