  The theory of greedy-like bases started in 1999 when S. V. Konyagin and V. N.
Temlyakov introduced in \cite{KT} the famous Thresholding Greedy Algorithm.
Since this year, different greedy-like bases appeared in the literature, as for
instance: quasi-greedy, almost-greedy and greedy bases. The purpose of this
paper is to introduce some new characterizations of 1-greedy bases. Concretely,
given a basis $\mathcal B=(\mathbf x_n)_{n\in\mathbb N}$ in a Banach space
$\mathbb X$, we know that $\mathcal B$ is $C$-greedy with $C>0$ if $\Vert
f-\mathcal G_m(f)\Vert\leq C\sigma_m(f)$ for every $f\in\mathbb X$ and every
$m\in\mathbb N$, where $\sigma_m(f)$ is the best $m$th error in the
approximation for $f$, that is, $\sigma_m(f)=\inf_{y\in\mathbb{X} : \vert
\text{supp}(y)\vert\leq m}\Vert f-y\Vert$. Here, we focus our attention when
$C=1$ showing that a basis is 1-greedy if and only if $\Vert f-\mathcal
G_1(f)\Vert=\sigma_1(f)$ for every $f\in\mathbb X$.
