4

This might be a very dumb question but I am having a hard time to understand why dual of $l_1$ norm is $l_\infty$ and vice versa. The dual of a norm is denoted $\lVert\cdot\rVert_*$, defined as $$ \lVert z\rVert_*=\sup\left\{z^Tx\mid\: \lVert x\rVert\leq1\right\} $$ and $l_1$ norm is, $$ \lVert x\rVert_1 = \sum(|x_i|) $$ and $l_\infty$ norm is, $$ \lVert x\rVert_\infty = \max(|x_1|,...,|x_n|) $$ I am very confused on dual norm. Any explanation would be very helpful.

Thanks

Martin Sleziak
  • 56,060
user2104150
  • 179
  • Well, have you seen a proof that the sual of a dual norm is the original norm? – Cameron Buie Feb 14 '16 at 15:33
  • You get this via a rather direct computation, playing with basic notions like indicator function, etc. See my answer to a similar question: http://math.stackexchange.com/a/1477721/168758. – dohmatob Feb 15 '16 at 14:01
  • Are you working with these norms on $,mathbb R^n$? Since for sequences spaces $\ell_\infty^*\ne\ell_1$. – Martin Sleziak Feb 25 '16 at 14:31

0 Answers0