$PSPACE$ not equal $DSPACE(2^n) $
Can this be shown using the space hierarchy theorem?


Yes. By the space hierarchy theorem, there is a language $L\in\mathrm{DSPACE}(2^n)$ that is not in $\mathrm{DSPACE}(f(n))$ for any $f=o(2^n)$.  In particular, this language is not in $\mathrm{DSPACE}(n^k)$ for any&nbsp;$k$, so it is not in $\mathrm{PSPACE}$.


  Is that the most simple and straight-forward way?


It's hard to imagine anything simpler!