The first step is make sure that the Library has your correct email address. By going to your Online Account, you can see what email address the Library is using and if you need to, you can update it there as well.
If your email address is correct, the most likely problem is that the your email provider is treating Library notifications as spam. Sometimes you can catch this by checking your spam folder, but sometimes they won't show up at all. To ensure Library notices are not being blocked as spam, you can try the following:
Here are a few of the ways the third option is done.
Outlook, Hotmail, and Live Mail:
How to Manually add an Email Address to Your Safe Sender List:
If an existing email from the sender hasn't already been trapped in your junk folder, you can manually add him to your Safe List. Click on "Options" in the upper right corner of your Hotmail screen, then click the link for "Safe and Blocked Senders" in the body of the page under Junk e-mail. Click the "Safe Senders" link and enter the email address you want to white list. Click "Add" to put it on your list.
Gmail (Google Mail):