I am working my way through a book on functional analysis and I'm unsure how to approach this problem:
Suppose $A:E_1 \to E_2$ is a bounded linear operator between two normed linear spaces $E_1$ and $E_2$. Show $A$ is closed in the sense that if the following hold:
- $x_n \in D_A$
- $x_n \to x$
- $Ax_n \to y$
Then it must be true that
- $x \in D_A$
- $Ax = y$
Since the operator is between two normed linear spaces, then it follows that the operator is continuous since it is bounded (as boundedness and continuity is equivalent in spaces that satisfy the first axiom of countability).
I am unsure however of how far continuity can take me here. If $x \in D_A$ then continuity should imply (5), but how do I show that $x$ is necessarily in the domain of $A$?