Remark 2

Let E and F be two Banach spaces, and $(u_n)$ a sequence of continuous linear mappings from E into F such that $\sup \|u_n\| = + \infty$. Then the set X of all $x \in E$ such that $\sup \|u_n(x)\| = + \infty$ is dense in E and is the intersection of a sequence of open sets in E. For, let $X_k$ denote the set of all $x \in E$ such that $\sup \|u_n(x)\| > k$ (for k integer $> 0$). Each $X_k$ is open and X is the intersection of the $X_k$. Since E is a Baire space, it is enough to show that each $X_k$ is dense in E. But, if the complement of $X_k$ contains a non-empty open set U, we would have $\|u_n(x)\| \leq 2k$ for $x \in U - U$ and, since $U - U$ is a neighbourhood of 0, we would have $\sup \|u_n\| < + \infty$.