hanging swag fixture socket?

Have a "twin" swag lamp that is wired to into a wall fixture above a bathroom vanity and operates with a wall switch. Both lamps are supposed to turn on when the switch is flipped. One of the lamps doesn't work any more; the other works fine. I really like the fixture and don't want to replace it. What can I do to determine the problem with the non-working lamp and can I fix it myself? (and Yes, I've tried a new bulb!)
Debbie