Given a decision problem DEC which is PSPACE-Hard and a function problem FUN.  If there is a polytime reduction from DEC to FUN, does this mean that FUN is FPSPACE-Hard?In my case, the answer of the function problem is the one of the decision problem plus additional information.  So it seems natural to assume that it is at least as hard.  The question is whether this means that a lower bound for its complexity is the function class corresponding to the complexity class of the decision problem.
Complexity class when reducing decision problem to function problem
cc.complexity theory;reductions