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We know that $\mathsf{DSPACE}(\log\log n) = \mathsf{DSPACE}(1)$ according to this proof. Can we claim that $\mathsf{DSPACE}(n)=\mathsf{DSPACE}(n/\log\log n)$ or something like $\mathsf{DSPACE}(n^3)=\mathsf{DSPACE}(n^3/(\log\log n)^k)$ for some $k\in \mathbb{N} $?

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The space hierarchy theorem shows that your classes are different.

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