Let $(X, \| \cdot \|_X )$ and $(Y, \|\cdot \|_Y )$ be Banach spaces s.t. $Y \subset X$. Suppose that $Y$ is $\|\cdot\|_X$-dense in $X$ and that a sequence $\{L_n\}_{n \ge 1} \subset X'$ is given s.t. $$ L_ny \to Ly \quad \forall \, y \in Y,$$ where $L \in Y'$.
Can we prove that the $X$-operator norm of the sequence is uniformly bounded, i.e. that $$ \sup_{n \ge 1} \sup_{x \in X, \|x\|=1} L_nx< + \infty?$$
Such a sequence need not be bounded in $X'$. A simple counterexample can be found for $X = c_0$, with the supremum norm, and $Y = \ell^1$. Take $$L_n(x) = \sum_{k = 1}^n x_k\,.$$ It is easy to see that $\lVert L_n\rVert_{X'} = n$, thus the sequence is surely not bounded in $X'$. But for $y \in Y$ we have $$L_ny \to Ly = \sum_{k = 1}^{\infty} y_k$$ and $\lVert L\rVert_{Y'} = 1$.