1. Set a rack in the middle level of the oven and preheat to 350ºF (180ºC) degrees.
2. Stir together the flour, baking powder, baking soda, and cinnamon.
3. In a medium mixing bowl, whisk the egg whites to break them up, then whisk in the sugar and the brown sugar. Whisk in the butter and vanilla, followed by the mashed bananas.
4. Sift the flour mixture over the banana mixture and thoroughly fold it in. Scrape the batter into the prepared pan and smooth the top.
5. Bake the cake for about 55 to 65 minutes, or until a toothpick inserted in the center emerges clean.
6. Cool the cake in the pan on a rack for 10 minutes, then unmold it and cool completely on a rack.