how to prove that the operator T on $l^{2}$ is bounded if $$ T\{x_{n}\}=\left\{x_{1},\frac{x_{2}+x_{3}}{2!}, \frac{x_{4}+x_{5}+x_{6}}{3!}, \frac{x_{7}+x_{8}+x_{9}+x_{10}}{4!},\ldots \right\}. $$ Is it possible to factor out $\sum_{n=1}^{\infty }\frac{1}{n!}$ when counting the norm $||T\{x_{n}\}||$? I don't have much profficiency in sequences, so I'd greatly appreciate any advice. Thanks.
2026-04-13 19:55:27.1776110127
Prove that an operator is bounded
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Hint: Perhaps surprisingly (given your thoughts so far), we have $\|T\|=1$. Note that $$ |x_1 + x_2 + \cdots + x_n|^2 \leq n \left(|x_1|^2 + \cdots + |x_n|^2\right). $$