Hi Guys I am terribly stuck on this question.
$u_n$ are an unconditional but not shrinking, norm 1 basis of a banach space $Y$, which is the closure of the span of $u_n$.
$l_1$ is the sapce of sequences with the sum of their absolute values of all elements being finite.
$T$ is defined to be $T\left(c\right)=\sum_{n \in N} c_nu_n$ where $c=$$\left(c_n\right)_{n \in N}$ is a sequence in $l_1$.
How do I show that T is an isomorphism between $l_1$ and $Y$?
I argued that as Y is the closure of span, every linear combination of $u_n$ is in Y, and as $u_n$ is a basis the mapping is injective from $l_1$ to $Y$. (I am not sure if that is right though). But I have no idea how to show surjectivity and that the mapping is bounded.
Can anyone please provide some hint?