Let $(V_1, \|\cdot\|_1)$ be a Banach space, $(V_2, \|\cdot \|_2)$ be a normed space. Let $(T_n)_n \subset B(V_1,V_2)$. Prove the eq. of the following statements.

a) If $\sum_{n=1}^{\infty} x_n$ is convergent in $V_1$, then $T_nx_n$ converges to $0$ in $V_2$.

b) $\sup_n \|T_n\| < \infty$

I thought I start with $a \to b$.

I figured if I could prove that $\sup_n \|T_nx\| < \infty$ .i.e pointwise bounded. Then result follows from Uniform boundedness. How can I from the information in a) draw the conclusion that all operators are pointwise bounded?


It will be easier to prove the contrapositive. Assume we have $x\in V_1$ such that $\sup_n\|T_nx\|=\infty$, so there exists a subsequence $\{n_k\}_{k\ge1}$ such that $\|T_{n_k}x\|\ge 2^k$. Define

$$x_i:=\begin{cases}2^{-k}x&\text{if }i=n_k,\\ 0&\text{otherwise.} \end{cases}$$

Clearly $\sum_{n=1}^\infty x_n=x$, but $\|T_{n_k}x_{n_k}\|\ge1$ for every $k\ge1$, so in particular $T_nx_n\not\to0$. This shows $a\to b$, and the other direction is trivial.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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