Let $T$ be a linear transformation, $T: c_{00} \rightarrow \Bbb C$
I want to show that if $c_{00}$ is equipped with the norm $||\bullet||_2$ from sequence space $l^2$, then T is not a bounded linear transformation. I'm relatively new to functional analysis, so would appreciate some guidance. In order to prove this, am I correct in thinking that I need to find a counterexample, i.e find a sequence $(x_n)_{n \in \Bbb N}$ with finite non-zero terms and then show how with T it is unbounded?
I'm stuck with finding such a counterexample, could someone point me in the right direction? TIA
First of all, you need slightly more from the sequence. Examine the definition of the operator norm. A linear map is bounded if and only if the operator norm $$ ||T||:=\sup\{||Tx|| : x\in c_{00}, ||x||\leq 1\} $$ is finite. So to prove that $T$ is unbounded, you need to show that that there is a sequence $(x_n)$ with $||x_n||\leq 1$ for all $n$ such that $||Tx_n||>n$. (As an exercise, prove that the operator norm is not finite if and only if this condition holds.)
In your case, you need to know something about $T$ in order to show that it is unbounded; it is not true that all linear maps from $c_{00}$ are unbounded.