Sequences and bounded injective linear operator Let $X$ be a Banach space and $L:X\to X$ be a bounded linear operator which is one-to-one and not surjective, but with dense range. Suppose $(x_n)$ is a sequence of elements of $X$ such that the sequence $(Lx_n)$ converges to 0. Does it follow that $(x_n)$ converges to $0$?  

Let $X = \ell^2$ and $L$ be given by 
$$L (y_1, y_2, \cdots, y_n , \cdots) = (y_1, y_2/2, \cdots, y_n/n, \cdots).$$
$L$ is bounded, one to one and not surjective (since the inverse $L^{-1}$, if exists, would have to be unbounded, and that would violate bounded inverse theorem) and has dense range (think of $L(n e_n) = e_n$). Note $L(e_n) \to 0$, but $e_n$ does not tend to $0$. 
