Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $T_n$ be a sequence of continuous linear operators from a Banach space $X$ to a normed linear space $Y$. Now, for all $x \in X$, $\lim_{n \rightarrow \infty} T_n(x)$ exists in $Y$. Show that the sequence $T_n$ is uniformly bounded.

I can prove that if we define $T(x) = \lim_{n \rightarrow \infty} T_n(x)$ for each $x \in X$, then $T$ is bounded. The problem is I don't know where to start when proving this proposition.

share|improve this question
add comment

2 Answers 2

You may want to use the principle of uniform boundedness.

Since the limit of $T_n(x)$ exists for all $x\in X$, $\|T_n(x)\|\le C_x$ must be bounded for all $x\in X$ (note that the bound depends on $x$, but not on $n$). The PUB now states that $\|T_n\|$ is bounded.

As Davide Giraudo already noted, the PUB is a consequence of Baires category theorem.

share|improve this answer
add comment

Use Baire categories theorem with the closed sets $$F_N:=\bigcap_{n\in\Bbb N}\{x,\lVert T_nx\rVert\leqslant N\}.$$

What we prove is the principle of uniform boundedness.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.