I'm looking for a vector space $V$ and a series $(T_n)_{n \in \mathbb{N}}$ in $L(V,V)$ such that $T_n$ converges pointwise to zero, but the operator nom $||T_n||$ doesn't converge.
2026-04-02 19:13:29.1775157209
Example of sequence of continuous linear maps converging pointwise to zero without converging in norm
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Consider $T_n:l_p\to l_p$, $1\leq p<\infty$, $T_nx=(0,0,...,0,x_n,0,0,...)$. This sequence converges pointwise to the zero operator, but does not converge to the zero operator in the norm.