If you see your opponents hand, is it legal to write down the cards so you remember them?

Yes. Any time you see anything while playing an opponent, you may have a pad of paper and pen to record what cards they have. This includes graveyards, revealed hands, and any effect that allowed you to see there card. You may also record life points, counters, names, and the DCI number of your opponent. You will actually see a lot of this happen in tournaments. The only words of caution I give you is if you do this, do not slow down the game in any way. If you do, an opponent may call a judge over and you will get a warning for slow play. If this happens again, you will get a game loss. The other rule is if you are walking around between matches while other matches are happening, you may observe, but you can not right down information you are watching. You can only write down information that is happening during your current match.

Thank you kindly good sir.