negation - What is the difference between "there's no" and "there's not"? When to use "not" and when to use "no"?

As we know, not and no are different words, which is why I don't know if there is a difference between "there's no" and "there's not".

This diagram shows that the two forms are used and this makes me think that they are both acceptable.

My question is: Are they both grammatically correct? If so, is there any rule regarding the use of one over the other, lest we should use them wrongly?

Answer

Forget "no" and "not" for a second - they seem related, but it is a false comparison.

Instead, think about

"There is" vs. "There is not". These are related expressions but one is positive and one is negative.

Things are simple when the (reflexive) subject of the sentence is expressed positively.
For example, suppose the subject is "life".

There is (life on Mars).

There is not (life on Mars).

The complication is that sometimes the subject is expressed negatively.
Suppose the subject was "no life".

The verb-positive sentence remains the same.

There is (no life on Mars).

The verb-negative sentence has to change, though, because English (unlike Spanish) has rules against double negatives.

It is incorrect to say:

There is not (no life on Mars).

Instead, we say

There is not (any life on Mars).

Other examples of a negative subject becoming positive because the verb is negative:

There is no reason to agree with him. There is not any reason to agree with him.

There is no way out of this situation. There is not a way out of this situation.

There is no one alive who remembers him. There is not anyone alive who remembers him.

In each case, the "not" in "There is not" is part of the verb, making the verb negative, whereas the "no" is part of the subject, making the subject negative. So you can't really compare "there is no" with "there is not".