TITLE: Application of uniform boundedness principle
QUESTION [5 upvotes]: Let $(a_n)$ be a sequence in $\mathbb{K}$ such that for each $(x_n) \in c_0$ also $(a_nx_n) \in c_0$.
Derive from the uniform boundedness principle that $(a_n) \in l^\infty$.
I see that the idea is to find a family $\{T_i\}_{i\in I}$ of linear, bounded operators that is pointwise bounded, and such that I can derive the desired result from the uniform boundedness.
However, I can't figure out how to define my operators to achieve this goal. Can anyone drop me a hint?

REPLY [4 votes]: Try $T_n\colon c_0\to c_0$ defined by $$T_n(x)=(a_1x_1,\ldots,a_nx_n,0,0,\ldots).$$

REPLY [2 votes]: Let $A$ be the given sequence. Let $AC$ be the sequence obtained by pointwise multiplication, given a sequence $C$. Clearly $A$ defines a linear map on sequences. Our hypothesis says precisely that $A$ is actually a map from $c_0$ to $c_0$. We need a norm on $c_0$. The sup norm seems nice, so let's do that. Which operators should we look at? Well, I would suggest looking at operators that "cut off" the sequence $A$ and make it zero after a certain point.
EDIT: Adding some more details, if $1_N$ is the sequence that is N ones followed by all zeros, then the sequences $A1_N$ obviously define a family bounded linear operator on $c_0$ with the sup norm, and you can show $\sup_{1 \leq n \leq N} |a_i| \leq \|A1_N\|$.