I'm studying functional analysis and I don't know how to solve the following problem:
Let $X$ be a normed space and $T:l^p\rightarrow X$ a linear continuous operator. Show that $||T||\geq \sup_{n\in\mathbb{N}}\{||Te_n||\}$. Fow which $p$ does the inequality becomes an equality?
I have easily shown that $||T||\geq \sup_{n\in\mathbb{N}}\{||Te_n||\}$. For the second part I know I have to find a condition on $p$ such that $||Tx||\leq \sup_{n\in\mathbb{N}}\{||Te_n||\}||x||$, but I don't know how to do it. Can anyone help me?
Thanks in advance.
The equality holds for $p=1$ for every $Y.$ Indeed $x=\sum_{n=1}^\infty x_ne_n$ is convergent. Hence $$\|Tx\|=\left \|\sum_{n=1}^\infty x_nTe_n\right \|\le \sum_{n=1}^\infty |x_n|\|Te_n\|\\ \le \sup_n\|Te_n\|\,\|x\|_1$$ The equality does not hold for $p>1.$ Indeed let $Y=\mathbb{R}$ and $Tx=x_1+x_2.$ Then $\|Te_1\|=\|Te_2\|=1,$ $\|Te_n\|=0$ for $n\ge 3,$ and $\|T(e_1+e_2)\|=2.$ Hence $$\|T\|\ge {\|T(e_1+e_2)\|\over \|e_1+e_2\|_p}={2\over 2^{1/p}}>1$$ Since $\mathbb{R}$ is the subspace of any normed space $Y,$ the equality for $p>1$ does not hold for any $Y.$