Let $X, Y$ be two Banach spaces and $T: X \to Y$ be a bounded linear operator. Suppose that $T$ is surjective. Given a sequence $y_k \in Y$ such that $y_k \to y_0$. Does there exists $x_k \in T^{-1}y_k$ such that $x_k \to x_0$?
In the case when $X$ is a Hilbert space, when I can decompose $X$ as $X = \mathrm{ker} T \oplus Z$ where $Z$ is the orthogomal complement of $\mathrm{ker} T$. Then $T: Z \to Y$ is an isomorphism, and we can easily obtain this result. However, for a general Banach space $X$, this doesn't work, according to this question.
