Banach-Steinhaus theory revisited: lineability and spaceability (Q346219)

Let \((T_n)\) be a sequence of bounded linear operators between Banach spaces \(B_1\) and \(B_2\). As a consequence of the Banach-Steinhaus theorem, if there is a vector \(f_1 \in B_1\) such that NEWLINE\[NEWLINE\lim \sup_{n \to \infty} \|T_nf_1 \| = \infty,NEWLINE\]NEWLINE then in fact the sequence \((T_n(f))\) diverges for a residual set of points \(f \in B_1.\) The authors' interest here is in the structure of this residual set.NEWLINENEWLINEThe main result is the following NEWLINENEWLINENEWLINETheorem: Assume that \((T_n)\) is a sequence of bounded linear operators between Banach spaces \(B_1\) and \(B_2,\) such that the following hold:{\parindent=6mm \begin{itemize}\item[(1)] \(\lim \sup_{n \to \infty} \|T_n\| = \infty\), and \item[(2)] there is a bounded linear operator \(T:B_1 \to B_2\) and a dense set \(\mathcal K\) such that \(\lim_{n \to \infty} \| T(f) - T_n(f) \| = 0\) for all \(f \in \mathcal K\). NEWLINENEWLINE\end{itemize}} Then the set \(D = \{ f \in B_1 \mid \lim \sup_{n \to \infty} \|T_nf\| = \infty \}\) is lineable. That is, \(D \cup \{0\}\) contains an infinite-dimensional vector space.NEWLINENEWLINEThe argument rests on the following NEWLINENEWLINENEWLINELemma: Under the assumptions of the Theorem, there exist a linearly independent sequence \((\varphi_n)\) contained in the unit ball of \(B_1\) and a constant \(C_L > 0\) such that, for all \(n \in \mathbb N\), there exists a sequence of natural numbers \((N_k (n))_{\{k\in \mathbb N\}}\) satisfying the following two conditions:{\parindent=6mm \begin{itemize}\item[(i)] \(\lim \sup_{k\to \infty} \|T_{N_k (n)} \varphi_n\| = \infty,\) and \item[(ii)] \(\sup_{k \in \mathbb N} \| T_{N_k (m)} \varphi_n \| \leq C_L\) for all \(m \neq n\). NEWLINENEWLINE\end{itemize}} An analogous theorem for spaceability is also presented.