# What is a standard name for this kind of unconditional bases in Banach spaces?

I am looking for a standard name (if it exists) for the following property of a Schauder basis $(e_i)_{i=1}^\infty$ in a Banach space $X$:

$$\|\sum_{i\in F}x_ie_i\|\le\|x\|$$for any $x=\sum_{i=1}^\infty x_ie_i\in X$ and any finite subset $F\subset\mathbb N$.

This condiion implies that the Schauder basis is unconditional. Can a Schauder basis with this property called monotone uncounditional basis? Or the latter term usually means something else?

The terminology I have seen in the literature refers to such a sequence as being a $1$-suppression unconditional basis. More generally, if for $K\geq1$ we have $$\Vert \sum_{i\in F}x_ie_i\Vert \leq K\Vert x\Vert$$ for every $x=\sum_{i=1}^\infty x_i\in X$ and finite $F\subset\mathbb{N}$, then we say that $(e_i)_{i=1}^\infty$ is $K$-supression unconditional. The least constant $K$ for which the above inequality holds is called the suppression constant of $(e_i)_{i=1}^\infty$, which is sometimes denoted by $K_s$. (If $K_u$ is the unconditional basis constant of $(e_i)_{i=1}^\infty$, then clearly $1\leq K_s\leq K_u\leq 2K_s$.)