I can't shut down the Mail application.

try force quiting mail

right click the icon for mail in your dock, it might take a second to pop up the quit option. Now when that window is up hold the option key down. Quit will change into force quit. Click force quit.

One more way if you wanted, >Force Quit> Selcet Mail> Force Quit

Thanks for your advice, it worked! Last night, a message wouldn't send, and I went to Safari, and found my internet connection was down. I could not 'force quit mail' as the option was greyed out...thismorning the internet connection was back, but I still could not force quit mail-and the Apple store phone was closed..so I came on to this community [i am new to computers],but I could understand your answer.

thanks i got it to work

I had this issue and got frustrated having to force quit all the time.

I usually quit all programs with shortcut keys.

MAIL now quits everytime if I just click Mail-Quit Mail

Don't use shortcuts to quit it and it will be fine