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I think that a commont common technique for proving such statements is for example the following type:

On

One class shares a closure property, while the other cannot because of a hierarchy theorem. Thus the they cannot be equal.

In this particular case a proof should could proceed along these lines: Since NP is closed under polynomial time reductions, so would SPACE(n), if they were equal. Then deduce that polynomial time reductions would imply that SPACE(n^2) is contained in SPACE(n), which is impossible by the spache space hierarchy theorem.
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I think that a commont technique for proving such statements is for example the following type:

On class shares a closure property, while the other cannot because of a hierarchy theorem. Thus the cannot be equal.

In this particular case a proof should proceed along these lines: Since NP is closed under polynomial time reductions, so would SPACE(n), if they were equal. Then deduce that polynomial time reductions would imply that SPACE(n^2) is contained in SPACE(n), which is impossible by the spache hierarchy theorem.