What should a polite reminder email look like?
While the comment of Solar Mike is correct (patience), and you don't want to be a pest, there is one thing you can do that would be useful in any case. 

Send a mail asking for any help on getting an early start. Are there papers you should read, for example? That keeps you in the mind of the professor, advances your cause, but also gives you something to do that can increase your knowledge. 

But "nearly a week" is a nano-second in academic time.