Social networks are notorious for sending a huge amount of emails, and depending on the settings you have enabled on your account, Facebook can be one of the the worst culprits on this. It's possible that you get a notification for every friend request, tagged picture, and even like that your account gets, which adds up fast.
The first thing you should do is to go to your Facebook account directly and try to disable the notifications from there. <a target="_blank" href="https://www.facebook.com/help/154884887910599?helpref=faq_content">Here</a> is a nice guide from Facebook directly showing how to do that.
However, for whatever reason you might not be able to access the account anymore, or maybe disabling the notifications via Facebook still doesn't get them to stop. If that's the case, then you can prevent the emails from reaching your inbox using Block Sender.
Follow these steps to get rid of unwanted Facebook emails:
<ul>
<li>Install the Block Sender browser extension (<a target="_blank" href="https://help.blocksender.io/installing-block-sender-in-chrome">Chrome</a>, <a target="_blank" href="https://help.blocksender.io/installing-block-sender-in-firefox">Firefox</a>), or just <a target="_blank" href="https://help.blocksender.io/how-to-create-a-block-sender-account">sign up</a> for an account, and get your account setup.</li>
<li>Open your Gmail account, find a notification from Facebook, and open the email</li>
<li>Click the &quot;Block&quot; button in the toolbar above the email and select &quot;Emails from this sender&quot;</li>
</ul>
And that's it! You'll no longer receive emails from that Facebook email address. Note that Facebook uses many different email addresses to send email, so you might be better off blocking the entire domain, like facebookmail.com or support.facebook.com, instead of the specific email address.
Important: Be careful with this since you may also block actual important emails from Facebook like security notifications, password reset emails, etc.

How to Stop Spam Emails from Facebook