What's means ''k-unconditional basis''?

Question

I'm studyng Schauder basis in Banach spaces and I don't find a definition of the "k-unconditional basis".

In the book "Classical Banach spaces I - Lindenstrauss and Tzafriri" the authors define "basis constant" and "unconditional constant" (pág. 18 and 19).

In the expression "Tsirelson space has a 1-unconditional basis" the constant "1" is relative to "unconditional constant"? or "basis constant"?

Thanks for any help!

Answer 1

Carothers p. 30 proves the

Theorem: a sequence $(x_n)$ of nonzero vectors is a basis for the Banach space $X$ if and only if $(x_n)$ has dense linear span in $X$ and there is a constant $K$ such that $\left \|\sum^n_{i=1}a_ix_i\right \|\le K\left \|\sum^m_{i=1}a_ix_i\right \|$ for all $a_i$ and $m>n.$

From this, the $\inf$ over all such $K$ seems to fit the definition.