Performance anxiety is a real thing - it is a huge obstacle to many pianists. It also varies with personality. An outgoing person that likes human interaction seems to be less affected by it. An introvert or someone who is less interested in personal interaction are more susceptible to it.
If you can expose yourself to public performance earlier on in your life, chance are, you can adapt to it easier. There are many ways you can improve on your performance anxiety.
a. Give yourself opportunity to perform in public - Start in smaller venue. You can start by performing for a few friends. Local senior center is also a great place to perform as you get to play in front of people who are supportive.
b. Breathing techniques - Deep breathing technique helps. Do you know deep cleansing breathing can help relieve stress? Breathing technique is common used in labor room. Once you take deep breath you tend to forget about what you are focusing on.
c. Know your music - This may seem like a no brain er. Many performers may not know their music as well as they thought. Are you able to pick up any measure without starting from the beginning? A good way to practice and memorize music is by breaking pieces into many smaller sections. If you can start the music anywhere on the piece, your chances of forgetting (the music) are much slimmer.
d. Take it less seriously - Don't think it is a big deal to mess up the performance (really)! Just have fun and enjoy the process. If you have the correct mentality and attitude, you will less likely be overly anxious and nervous.