I'm reading a paper and having some trouble with a certain inequality.
Let $W$, $X$, and $Y$ be Banach spaces, with $(x_n)$ a normalized basic sequence in $X$ and $(w_n)$ a normalized unconditional basis in $W$. Let $T:X\to Y$ be a bounded linear operator and let $\delta>0$. Suppose there exist matching block basic sequences $u_n=\sum_{k\in I_n}a_kx_k$ and $v_n=\sum_{k\in I_n}a_kw_k$ such that $\|Tu_n\|<\delta_n\|v_n\|$ for all $n$, where we define $\delta_n=\delta 2^{-n}$.
Claim of the paper: For any finite sequence of scalars $(b_n)$, we have $\|\sum b_n Tu_n\|<\delta\|\sum b_nv_n\|$.
What is the justification for this claim?
Additionally, I am curious about something else, which is not part of the paper I am reading. Here is the object of my curiosity: Suppose $(z_n)$ is a normalized basis for a Banach space $Z$ and that $S:Z\to Z'$ is a bounded linear operator between Banach spaces. Let $C=\sup\{\|Sz_n\|:n\in\mathbb{N}\}$. What is a sufficient further assumption for $(z_n)$ to guarantee that $\|S\|=C$? If $(z_n)$ need only be unconditional, then the above claim is immediate. But that seems too good to be true, so probably more is required than just unconditionality.
Thanks guys!