I've a problem where I have to prove the following statements:
(i) if $SPACE(n) \subseteq P \implies SPACE(n^2) \subseteq P$
(ii) if $P = SPACE(n) \implies SPACE(n) = SPACE(n^2)$
For the Space Hierarchy Theorems we have that $\forall f(n) = o(g(n)) \implies SPACE(g(n)) \nsubseteq SPACE(f(n))$
Using this result and the statements (i) and (ii) I have to prove that (iii) $P \neq SPACE(n)$.
I'm using the following post as reference: https://mathoverflow.net/questions/40770/how-do-we-know-that-p-linspace-without-knowing-if-one-is-a-subset-of-the-othe/333956#333956
I suppose that I can use Thomas Klimpel's proof in the abovementioned link to prove the statement (i). But how can I prove the statements (ii) and (iii)?
Accept my apologies for this question, but I'm a rookie in Theory of Complexity :)