I have come across the following construction, but l am not sure whether it is correct or not.
Not a lot of his friends speak English well.
My question is: can we use "a lot of" in a negative sentence as mentioned above?
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Yes, you can use this construction. From the Merriam-Webster dictionary:
Definition of a lot of
a large number or amount of (things, people, etc.)
A lot of people feel that way. There was a lot of space. Not a lot of people know that.
I think it's fine for informal usage, but I personally would avoid using it in formal, written English. The word lot, defined as a large number or amount, often carries the "informal" label in dictionaries. Not a lot of seems even more informal to me.