Is it true that P is not equal to deterministic linear space complexity class?
We do know that $\mathsf{P} \neq \mathsf{DSPACE}(n)$. This is proved in answers to a question on Mathoverflow. The idea is that $\mathsf{P}$ is closed under polynomial blowup while $\mathsf{DSPACE}(n)$ isn't (due to the space hierarchy theorem); consult the link on Mathoverflow for the details.