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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 $\mathbb R^n$

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    In finite dimensions (where stating $n$ makes sense) all norms are equivalent. – 2017-02-18
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    So in an infinite dimensional vector space $l^2$ and $l^3$ norms are not equivalent. Am I right ? – 2017-02-18
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    What does "equivalent" mean when the spaces are different? Say, $(1,1/\sqrt2,\ldots,1/\sqrt n,\ldots )$ is in $\ell^3$ but not in $\ell^2$. – 2017-02-18
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    Maybe take $\ell^2$ as subspace of $\ell^3$. – 2017-02-18
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    You already have an answer to the question as stated now (all norms in $\mathbb R^n$ are equivalent). So change the question if you are asking for something else. – 2017-02-18

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