A banana, or any food containing lots of carbohydrates (e.g. fruits, bread, cereal) are a terrible thing to have first thing in the morning, unless it's AFTER your workout. Here's why:
1. The human body stores (in muscles and liver) around 15g of glucose per kg of bodyweight. It does so in the form of glycogen. So if you weight 70kg, you have around 1,000 of glucose already stored and ready for use. That's equivalent to 4000 Calories.
When you wake up in the morning, those stores are still almost full (you haven't used much them while sleeping except a bit by the brain), so any additional glucose you eat (or drink) will be converted to fat, because there is no more storage space to put it in.
If you work out before breakfast, you use up some of that stored glucose, so the carbs you eat after your workout will be stored in the muscles instead of being converted to fat (don't overeat!).
2. When you consume any carbohydrate, your body produces Insulin. Insulin shuts down the action of an enzyme called Hormone Sensitive Lipase, which is responsible for breaking down fat for energy.
So when this enzyme is shut down, your body's ability to burn fat is significantly reduced (both during a workout as well as during the day).
I almost never have carbs before lunch or even dinner. The only exception: when I have a double workout day and I need to recover after my first workout.