# Tape reduction, tape compression and time compression

In our lecture we have the following relationships: I have problems to understand these abstract classes. First of all, our Turingmachines are defined as $$1$$ input tape and $$k$$ working tapes. DSPACE(f)(resp. NSPACE(f)) means all languages that are excepted by Turingmachines which are deterministic (resp. non-deterministic) and f-spacebounded. DTIME(f)(resp. NTIME(f)) means all languages that are excepted by Turingmachines which are deterministic (resp. non-deterministic) and f-timebounded.

If I look at $$1$$ for DSPACE, I get $$DSPACE(\mathcal{O}(f)) = DSPACE_{1-tape}(f).$$ So I know that the left side is the union of all DSPACE($$c\cdot f), c\in\mathbb N$$, but no amount of tapes is given. So, does that mean, if I have a problem and I know it can be solved in time $$15^{7}\cdot f$$ and needs a large number of tapes, then I know there is a TM that solves the problem in time $$f$$ and needs only $$1$$ tape?

Point $$2$$ is even more confusing. I understand it in the way that it says, if I have a problem that needs time $$15^7\cdot f$$, then there is a TM for the same problem that only needs time $$f$$. I am am consufed because the folliwng theorem (Hennie-Stearns) in the lecture says but here I have a number $$k$$ of tapes?