A time-space hierarchy between polynomial time and polynomial space

In this paper we define a natural time-space hierarchy $$P = F_0 subseteq F_1 subseteq F_2 subseteq cdot cdot cdot subseteq PSPACE$$ between polynomial time and polynomial space and then study the relativizations of this hierarchy, obtaining a relative collapse and a relative proper hierarchy, at each possible level. Some partial results concerning relativized time-space tradeoff are given, together with some open questions.