When does convergence in range imply convergence of arguments? Let $H$ be a Hilbert space and $T : H \rightarrow \mathbb{R}$ be bounded and linear, and suppose that we have
$$\tag{1}\lim_{n\rightarrow\infty} T(x_n) = y$$
for a sequence $(x_n)$ in $H$ and $y\in\mathbb{R}$. Do you know of a (minimal) sufficient condition on $T$ which allows us to conclude that
$$\tag{2}x_n\rightarrow x \quad\text{ for some }\quad x\in H \ ?$$
Remark: Obviously the above fails if $T=0$ (say), and it holds if $T$ is a bijection (trivially in this case, and by the bounded inverse theorem for more general codomains); but is $T$ bijective also necessary for $\big[(1) \Rightarrow (2)\big]$ to hold?
 A: If $T$ is not injective, then choosing distinct points $a$ and $b$ in its kernel and defining a sequence $x_n$ in $H$ by the recipe $x_n = a$ if $n$ odd and $x_n = b$ if $n$ even will produce a non-convergent sequence $x_n$ in $H$ for which the sequence $Tx_n = 0$ is convergent.  So it is necessary that $T$ be injective for the desired condition to hold.  And if $T$ is injective, it is nonzero (outside of the trivial case $H = \{0\}$ which can be handled separately).  And a nonzero linear functional is surjective.  Hence $T$ is a bijection, where the argument you hint at applies.
If the codomain of $T$ is not assumed to be $\mathbb{R}$ but just a more general complete linear space (i.e., one where "$T$ nonzero" does not automatically imply "$T$ is surjective"), it appears to be necessary that $T$ be both injective and have closed range.  If the range of $T$ is closed then your "bounded inverse theorem for more general codomains" (you don't state the hypotheses, but I could guess them) likely applies.  If the range of $T$ is not closed, then there is a sequence $x_n$ in the unit sphere for which $T x_n \to 0$, see e.g. Closed Range (injective operator), and if the sequence $x_n$ happened to be norm convergent, you could multiply it by a sequence of unimodular scalars (e.g. $(-1)^n$) to produce examples of sequences that do not converge with the same property.
