Preheat oven to 350 degrees Fahrenheit (175 degrees Celsius).

Lightly grease a 9x5 inch loaf pan with oil.

In a bowl, combine cake flour, baking powder, baking soda, and ground cinnamon.

In a separate bowl, whisk together applesauce, granulated sugar, egg whites, mashed bananas, and vanilla extract.

Add the flour mixture to the wet ingredients all at once.

Stir gently to blend the ingredients, ensuring not to overbeat.

Pour the batter into the prepared loaf pan.

Bake for 45 to 50 minutes, or until a knife inserted into the center comes out clean.

Let the banana bread cool completely in the pan before slicing and serving.