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If $1\le p\le q\le \infty$, we know that the following inequality holds:

$$\|a\|_q\le \|a\|_p.$$

What could be a possible interpretation of this inequality for a non-mathematician? For example, can we say something like "the $l_p$ norm becomes more robust (or sensitive) to outlying values with the increase of $p$"?

Shanks
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    Note that this is not true if you are referring to $L^p$ (with capital $L$; ie for real-valued functions); it is only true with $l^p$ (with lowercase $l$; ie for say sequences). – HK Tan Jul 03 '22 at 06:01

1 Answers1

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The inequality $\lVert a\rVert_q\leqslant \lVert a\rVert_p$ means that if $\lVert a\rVert_p\leqslant 1$, then $\lVert a\rVert_q\leqslant 1$ or in other words, that the unit ball for the $\ell^p$-norm is contained in the unit ball for the $\ell^q$-norm.

For an interpretation of the unit ball of these spaces, you can have a look here.

Davide Giraudo
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