Don't banks fill an important role by allowing people to take out loans and so increasing people's capacity to invest in their businesses, buy houses, etc? How will Bitcoin replace that role if there's no third-party to store people's money, deal out loans and charge and distribute interest?
I'm sure this has already been covered somewhere but I haven't come across the answer yet.
I'm not sure that lots of the answers here address the question.
If you are asking, how can a bank worth without fractional reserve banking, I would say that is can't really, or at least it would be very different.
A bank has no incentive to lend 500BTC to someone to buy a house, when they will only receive 500BTC back. Given that sometime inflation of the BTC supply will be zero, they could not charge interest as the money simply wouldn't exist if this was repeated around the world. As fiat money at the moment is debt, it works with the fractional reserve system, if the whole world used Bitcoin, it wouldn't.
Islamic banks as far as I understand would work with Bitcoin, maybe that would be the future..