Today's library is the internet. Storage, transfer, and duplication costs are so much lower for digital works that we're able to have a huge amount of information available to all. You can't get everything online yet, but this continues to get better and the internet is already well past the best physical libraries. Payment is tricky, with a huge gap between free/ad-supported and paid, but I don't think this is permanent.

So where does this put public libraries? Their main advantage, centralization, is gone. They still lend books [1] but we're getting more and more online and e-readers are getting better and cheaper. They seem to have taken on two new functions: (1) providing computers and an internet connection and (2) paying for their members to get access to ebooks, journals, and databases. When we decide that storing all those paper books redundantly in buildings across the country is too expensive, will libraries continue just in these two new functions?

[1] Which the internet improves: I can request books online and get an email when they're ready to be picked up at the branch library I walk past on my way home from work.