1
$\begingroup$

Given $p\in (1,\infty)$. Take a bounded sequence $(f_n)$ in $\ell^p$ and define a linear map $T\colon \ell^p_{00}\to \ell^p$ ($\ell^p_{00}$ is the space of finitelty supported sequences in $\ell^p$) by

$Te_n = f_n$

($e_n$ is the usual (Hamel) basis for $\ell^p_{00}$).

Is $T$ a bounded linear map? If $p=1$, the answer is yes... If not, what do have to assume on $(f_n)$ to get boundeness of $T$?

  • 0
    One nontrivial sufficient condition is the Schur Test ($p=2$), which you can find, for example, in Conway's *Course in Functional Analysis*. [The Wikipedia article states it in the integral form only.]2012-05-24

0 Answers 0