Meagre set and sequence of operators in Banach Space

Question

Let $X$ a Banach Space , $Y$ a Normed Space and $(T_n)_{n=0}^\infty$ a sequence in $L(X,Y)$ such that $\sup_n {\| T_n (x) \|} = \infty$

Show that:

• $ Z=\{ x \in X |{ \sup_n\|T_n (x) \|} \leq \infty\} $ is meagre
• $Z^c$ is dense in X and is a intersection of dense open sets

Edit: With a hint I changed the notation for some more clear and asked more clearly that is a question. This exercise a professor of Functional Analysis created for us students, but I have some difficult in Analysis and need a little help to start in this beginning of the course.

Thanks in advance

Answer 1

What you are actually trying to show is the Banach-Steinhaus theorem. The first step consists in writing $Z = \bigcup_{M\in \mathbb{N}} Z_M$, with $Z_M = \{x\in X| \sup_n ||T_n(x)|| < M\}$. By contradiction, if $Z$ is not meagre, there is a small ball in some $Z_M$. Thanks to this ball, you can try to show that actually $||T_n(x)||$ is bounded by some constant independant of $n$ on the unit ball and thus conclude.