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 \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 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, arXiv:1506.03397).


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