Let $X=(x_{ij})_{i,j=1}^\infty$ be the matrix such that for all $a=(a_n)\in l_p$ and all $k\in\mathbb{N}$ the series $\sum_{n=1}^\infty x_{kn}a_n$ converges in $\mathbb{F}$. We assume that the sequence $Xa=(Xa)_k=\sum_{n=1}^\infty x_{kn}a_n$ is an element of $l_p$. Thus $X:l_p\rightarrow l_p$ is a linear map.
Now fix $k,K\geq 1$. We define $X_k^K:l_p\rightarrow \mathbb{F}$ and $X_k:l_p\rightarrow \mathbb{F}$ by $$X_k^Ka=\sum_{n=1}^K x_{kn}a_n$$ and $$X_ka=\sum_{n=1}^\infty x_{kn}a_n.$$
- How do I show that $X_k^K$ is continuous?
- How do I show that $X_k$ is continuous?
- How do I show that $X$ is continuous?
What I know:
Continuity of a map $f$ means that $||fx||\leq C||x||$ for all $x$. So I want to show that $|\sum_{n=1}^K x_{kn}a_n|\leq C||x||=C\left(\sum_{n=1}^\infty |a_n|^p\right)^{1/p}$ and $|\sum_{n=1}^\infty x_{kn}a_n|\leq C\left(\sum_{n=1}^\infty |a_n|^p\right)^{1/p}$.
How do I do this?
Edit: So I think I have to use one of the consequences of either the Open Mapping Theorem, the Closed Graph Theorem or the Uniform Boundedness Principle, but I cannot see which one.
For the first one I would say: $$|X(a)|=|\sum\limits_{k=1}^{N} a_{nk} x_k| \leq \sum\limits_{k=1}^{N} |a_{nk} x_k| \leq (\sum\limits_{k=1}^{N} |a_{nk}|^p)^{1/p} (\sum\limits_{k=1}^{N} |x_k|^p)^{1/p}$$ (with the Hölder inequality)
and then we have $$\leq (\sum\limits_{k=1}^{N} |a_{nk}|^p)^{1/p} ||x||_p$$ and define $$ c= (\sum\limits_{k=1}^{N} |a_{nk}|^p)^{1/p}$$ since this sum is finite