showing a functional on $l^p$ is bounded

Question

let $l^p$ be a sequence space and $a: l^p \to \mathbb C$ be a linear functional defined by $x \mapsto \sum_{j=1}^\infty a_j x_j$. How do I show $a$ is a bounded? I want to show that there exists a fixed real constant $M>0$ s.t. for all $x\in l^p$ with $\|x\|_p = 1$, $|\sum_{j=1}^\infty a_j x_j| < M$.

Answer 1

No advanced theorems are needed.

Let $1<p\le \infty$ We are going to show that $\{a_n\}\in \ell^q,$ where $q$ is the dual exponent. Once it is done, the Hôlder inequality implies $\|T\|\le \|\{a_n\}\|_q.$

Assume by contradiction that $\sum |a_n|^q=\infty.$ Then there is an increasing sequence $n_k$ of positive integers so that $\sum_{j=n_{k-1}+1}^{n_k}|a_j|^q\ge 1.$ Let $$x_n={\overline{a_n}|a_n|^{q-2}\over \left(\sum_{j=n_{k-1}+1}^{n_k}|a_j|^q\right)^{1/p}}{1\over k}, \quad n_{k-1}<n\le n_k $$ following the convention $x_n=0$ if $a_n=0.$ Then $$\sum|x_n|^p=\sum {1\over k^p}<\infty$$ and $$\sum a_nx_n\ge \sum {1\over k}=\infty$$ Thus we have obtained a contradiction.

For $p=1$ we will show that $\{a_n\}$ is bounded, hence $\|T\|\le \|\{a_n\}\|_\infty.$ Assume by contradiction $|\{a_n\}\|_\infty=\infty.$ There is an increasing sequence $n_k$ so that $|a_{n_k}|\ge { k}.$ Let $x_n=\overline{a_n}|a_n|^{-1}{1\over k^2}$ for $n=n_k$ and $a_n=0$ for remaining values of $n.$ Then $\|\{x_n\}\|_1=\sum {1\over k^2}<\infty$ and $\sum a_nx_n\ge \sum {1\over k}=\infty.$

The same argument for $p=2$ has been applied here.

Answer 2

It is understood that the series $\sum a_ix_i$ converges to a complex number for every $x \in \ell^{p}$. Define $T_n(x)= \sum\limits_{k=1}^{n}a_kx_k$. Check that $T_N$ is a bounded linear functional with norm $(\sum\limits_{k=1}^{n}|a_k|^{q})^{1/q}$ where $\frac 1p+\frac 1q=1$. Now apply Uniform boundeness principle to conclude that $\sum |a_n|^{q} <\infty$ and use Holder's inequality to finish the proof.

EDIT: As suggested by Anne Bauval we can just use the fact that pointiise limit of sequence of bounded operators between Banach spaces is bounded: Another simple consequence of Uniform boundeness principle.