Are there problems that cannot be solved in polynomial space?

Question

There are problems that can't be solved in polynomial time, but are there problems that can't be solved using polynomial space?

Answer 1

Yes. In the extreme case, there are problems like the halting problem that are undecidable, so they can't be solved in any amount of space. On a more practical level, there's the space hierarchy theorem which can be used to construct a problem that can be solved in space O(2ⁿ) but not space o(2ⁿ), ruling out polynomial-space solutions.