Weight-almost greedy bases

We introduce the notion of a $\mathit{\text{weight-almost}}\mathit{\text{greedy}}$ basis and show that a basis for a real Banach space is $w$-almost greedy if and only if it is both quasi-greedy and $w$-democratic. We also introduce the notion of $\mathit{\text{weight-semi-greedy}}$ basis and show that a $w$-almost greedy basis is $w$-semi-greedy and that the converse holds if the Banach space has finite cotype.