My tutorial instructor told me today that the sequence space norms $l^2$ and $l^3$ are equivalent but the norms $l^1$ and $l^2$ are not equivalent norms ?
However I could prove that $\|x\|_2 \le \|x\|_1 \le \sqrt{n} \|x\|_2$. So the sequence space norms $l_1$ and $l_2$ are equivalent. Am I right ?
Moreover can somebody give me an example of two norms which are not equivalent ?
Edit 1:- $n$ is the dimension of the space. The space under consideration is $\mathbb R^n$