Being under no illusion refers to being fully aware of the facts of a particular subject. being under illusion refers to having misbelief about that particular subject. i hope this helps.

Hi, these are set phrases/expressions. The other answers are correct, but when you want to say that somebody has a mistaken belief you need to add the word "the" - so the sentence becomes "being under the illusion" (because it's a specific illusion, not illusion in general)

"Being under illusion" is having a mistaken belief about something. "Being under no illusion" is being fully aware of the true state of affairs.