A basis $(u_n)_{n=1}^\infty$ of a Banach space $X$ is unconditional if for each $x \in X$ the series $\sum_{n=1}^\infty u_n^*(x)u_n$ converges unconditionally. It is easily verified that $(u_n)_{n=1}^\infty$ is an unconditional basis of $X$ if and only if $(u_{\pi(n)})_{n=1}^\infty$ is a basis of $X$ for every permutation $\pi : \mathbb{N} \rightarrow \mathbb{N}.$
Q1 How to prove that a basis is unconditional if and only if there is a constant $C$ such that $\|P_{A^c}\| \leq C$ for every finite set $A \subset \mathbb{N}.$ This is problem no. 10.15 p.293 from book "Topics in Banach space theory by Fernando and Nigel j. Kalton"
From book "introduction to Banach spaces and their geometry by Bernard Beauzamy" For a Schauder basis $(e_n)_{n \in \mathbb{N}},$ the following properties are equivalent:
a ) $(e_n)_{n \in \mathbb{N}}$ is an unconditional basis.
b ) For every convergent series $\displaystyle \sum_{n \in \mathbb{N}} a_n e_n,$ and every sequence $(\epsilon_n)_{n \in \mathbb{N}}$ with $\epsilon_n = \pm{1} \forall n \in \mathbb{N},$ the series $\displaystyle \sum_n \epsilon_n a_n e_n$ converges.
c) For every convergent series $\displaystyle \sum_{n \in \mathbb{N}} a_n e_n,$ and every strictly increasing sequence $(n_i)_{i \in \mathbb{N}}$ of integers, the series $\sum_i a_{n_i} e_{n_i}$ converges.
d) For every convergent series $\displaystyle \sum_{n \in \mathbb{N}} a_n e_n,$ and every sequence of scalars $(b_n)_{n \in \mathbb{N}}$ such that, for all $n, |b_n| \leq |a_n|,$ the series $\displaystyle \sum_n b_n e_n$ converges.
e) For every $\epsilon >0,$ every convergent series $\displaystyle \sum_n a_n e_n = x,$ there is a finite subset $A_0$ of $\mathbb{N}$ such that, for every finite $A \supset A_0, \|\sum_{i \in A} a_i e_i - x\| < \epsilon.$
f) There exists a constant $K \geq 1$ such that if $A$ and $B$ are finite subsets of $\mathbb{N},$ with $A \subset B,$ then for any sequence $(a_n)_{n \in \mathbb{N}}$ of scalars : $\|\sum_{n \in A} a_n e_n\| \leq K \|\sum_{n \in B} a_n e_n\|.$
Q2 Does any of these items give this meaning that "a basis is unconditional if and only if there is a constant $C$ such that $\|P_{A^c}\| \leq C$ for every finite set $A \subset \mathbb{N}.$ Moreover, $K_{su} = \sup_{|A|<\infty} \|P_{A^c}\|.$"
Also, from book "Topics in Banach space theory by Fernando and Nigel j. Kalton" A basis $(u_n)_{n=1}^\infty$ of a Banach space $X$ is unconditional if and only if there is a constant $K \geq 1$ such that for all $N \in \mathbb{N},$ \begin{eqnarray} \| \sum_{n=1}^N a_n u_n\| \leq K \| \sum_{n=1}^N b_n u_n\|, \end{eqnarray} whenever $a_1,...,a_N,b_1,...,b_N$ are scalars satisfying $|a_n|\leq|b_n|$ for $n=1,...,N.$
Q3 What this mean?