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?



What is a standard name for this kind of unconditional bases in Banach spaces?
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 \begin{equation}\Vert \sum\_{i\in F}x\_ie\_i\Vert \leq K\Vert x\Vert\end{equation} 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 $1\leq K\_s\leq K\_u\leq 2K\_s$.)


As far as references go, most of the above is contained in [Remark 3.1.5](https://books.google.com/books?id=UG6zDAAAQBAJ&pg=PA53) of *Topics in Banach Space Theory* by Fernando Albiac and Nigel Kalton. See also the paper *Characterization of 1-almost greedy bases* by Albiac and Ansorena (doi: [10.1007/s13163-016-0204-3](https://doi.org/10.1007/s13163-016-0204-3), arXiv:[1506.03397](https://arxiv.org/abs/1506.03397)).


