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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?

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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.

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