In 1999 Konyagin and Temlyakov introduced the greedy algorithm and applied their method to seminormalized basis In Banach spaces. We make new approach for studying greedy approximations by getting rid of seminormalized condition. For this goal we do some modifications of basic definitions such as democracy, greediness and partial greediness. This method allows us to study unbounded basis as well. We reprove some known theorems using the new definitions. We proved that a basis is greedy if and only if it is unconditional and democratic, almost greedy basis is equivalent to being quasi greedy and democratic and partially greedy is being quasi greedy and conservative. We give an example to show the differences between the two methods. |
