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Hello all, I'm trying to find a good resource for a discussion on the relation between say, the p-norm of a vector (from a finite dimensional vector space) and its Euclidean norm. In my search on the internet and in various books, I only encounter basic, standard inequalities such as the Cauchy-Schwarz and Holder's inequality.

Are there textbooks that go more in depth than these two?

In particular, I'm interested in the following: if I have two unit vectors $\psi$ and $\phi$ (from $R^d$, say), that are $\epsilon$-close, meaning that $\|\psi - \phi\|_2 \leq \epsilon$, then what can one say about $\|\psi\|_p - \|\phi\|_p$? Intuitively, they must be close as well, but does the closeness depend on $d$, the dimension of the vector-space?

Any references or links or pointers would be greatly appreciated!

Thanks, Henry

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  • $\begingroup$ Is there a reason you ask about $\lVert\psi\rVert_p - \lVert\phi\rVert_p$ rather than the maybe more natural $\lVert\psi - \phi\rVert_p$? $\endgroup$
    – LSpice
    Commented Mar 29, 2011 at 4:32

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It is no better than what you get from Holder. Take the case where one of the vectors is zero.

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  • $\begingroup$ I have specified that $\psi$ and $\phi$ are unit vectors in the 2 norm, however. $\endgroup$
    – Henry Yuen
    Commented Mar 25, 2011 at 18:15
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    $\begingroup$ I missed that, but it doesn't matter much. Let one vector be $e_1$ and the other the normalization of $e_1$ plus a small multiple of $\sum_{k=2}^n e_k$. $\endgroup$
    – Bill Johnson
    Commented Mar 25, 2011 at 18:22

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