First, I assume that your e-mail client is sending the message without any error dialogs---and the that message is in the "sent" folder (or equivalent). If so, then we know that your SMTP settings are correct.
If you send to a bad address, the receiving server should send back a message saying that your message was undeliverable.
Have you tried sending a message to yourself?
Please give more detail--what version of Linux, what e-mail client, etc.