Math 6710 • Applied Linear Operators and Spectral Methods

For \(k \in \mathbb{N}\) let \(T_k : \ell^1 \to \ell^\infty\) be the linear operator given by

\[T_k(x_1, x_2, x_3, x_4, \ldots) = \sum_{j=1}^{k} x_j e_j + \sum_{j=k+1}^{\infty} x_k e_j\]


where \(e_j = (0, 0, \ldots, 1, \ldots, 0, 0, \ldots)\), i.e. it is the element of \(\ell^\infty\) with a single 1 in the
\(j\)th position and all other entries are zero.
\begin{enumerate}

\item[(i)]
Compute \(\|T_k\|\) for each \(k\), and conclude that each \(T_k\) is bounded.
\item[(ii)]
Prove that the sequence \((T_n)\) is strongly operator convergent.
\item[(iii)]
Prove that the sequence \((T_n)\) is not uniformly operator convergent.

\end{enumerate}