I'm new to complexity theory and am analyzing inclusions between complexity classes. Suppose we are given the following seven complexity classes
- $DTIME(n)$
- $DTIME(n^2)$
- $DTIME(2^n)$
- $DTIME(2^{2^n})$
- $SPACE(\log^2n)$
- $SPACE(n/\log n)$
- $SPACE(n)$
We have that
$DTIME(n)\subseteq SPACE(n) \subseteq NSPACE(n) \subseteq DTIME(2^n)$
and by the Time Hierarchy Theorem,
$DTIME(n)\subseteq DTIME(n^2) \subseteq DTIME(2^n) \subseteq DTIME(2^{2^n})$
So, my gut feeling is that
$SPACE(\log^2n) \subseteq DTIME(n) \subseteq DTIME(n^2) \subseteq SPACE(n/\log n) \subseteq SPACE(n) \subseteq DTIME(2^n) \subseteq DTIME(2^{2^n})$
However, I'm not sure where $SPACE(\log^2 n)$ and $SPACE(n/\log n)$ fit in. I think that this follows from the Space Hierachy Theorem -
$${SPACE}\left(o(f(n))\right) \subsetneq SPACE(f(n))$$
Is there another theorem or result I could review to justify my proposed order?