## On the existence of almost greedy bases in Banach spaces

### Volume 159 / 2003

Studia Mathematica 159 (2003), 67-101
MSC: Primary 46B15; Secondary 41A65, 46B20.
DOI: 10.4064/sm159-1-4

#### Abstract

We consider several greedy conditions for bases in Banach spaces that arise naturally in the study of the Thresholding Greedy Algorithm (TGA). In particular, we continue the study of
*almost greedy* bases begun in [3]. We show that almost greedy bases are essentially optimal for $n$-term approximation when the TGA is modified to include a Chebyshev approximation. We prove that if a Banach space $X$ has a basis and contains a complemented subspace with a symmetric basis and finite cotype then $X$ has an almost greedy basis. We show that $c_0$ is the only ${\mathcal L}_\infty $ space to have a quasi-greedy basis. The Banach spaces which contain almost greedy basic sequences are characterized.