It's impossible for a problem to require exponential space without being exponential-time.
- Consider that if an $EXPSPACE~~complete$ problem can be solved in $2^n$ time. It will now fall into the class $EXPTIME$.
- Then $EXPSPACE~~complete$ problems are in $EXP$ if they can be solved in $2^n$ time. This means they can reduce into $EXP~~complete$ problems and vice versa.
To me, this should be easy to write a proof that $EXPTIME$ = $EXPSPACE$.
My intuition tells me that if $Exptime$ = $Expspace$; then $PSPACE$ != $EXPTIME$,
Because $PSPACE$ already is not equal to $EXPSPACE$.
As an amateur, what would make this reasoning be wrong or right?