# Zoo of ideal Schauder basis

## TL;DR

This paper explores filter Schauder bases in Banach spaces, providing new examples beyond standard bases, and characterizing filters through these bases, thereby deepening understanding of their structure and relationships.

## Contribution

It introduces novel examples of filter Schauder bases in classical Banach spaces and characterizes filters via these bases, expanding the theoretical framework.

## Key findings

- New examples of filter Schauder bases in classical spaces
- Characterization of filters using Schauder bases
- Analysis of relationships between basic sequences and filters

## Abstract

We investigate the notion of filter (equivalently: ideal) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell_p$, $c_0$, James' space). Those examples lead to distinguishing and characterizing filters (equivalently: ideals) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and filters (equivalently: ideals) on the set of natural numbers.