Dual norm is defined as
‖z‖∗=sup{zTx|‖x‖≤1} The dual norm can be interpreted as the operator norm of zT, interpreted as a 1×n matrix with the norm ‖⋅‖ on Rn.
From the definition of dual norm we obtain the inequality
zTx≤‖x‖‖z‖∗
The conjugate of any norm f(x)=‖x‖ is the indicator function of the dual norm unit ball
f∗(y)={0‖y‖∗≤1∞otherwise
Proof. If ‖y‖∗>1, then by definition of the dual norm, there is a z∈Rn with ‖z‖≤1 and yTz>1. Taking x=tz and letting t→∞, we have
yTx−‖x‖=t(yTz−‖z‖)→∞ which shows that f∗(y)=∞. Conversely, if ‖y‖∗≤1, then we have yTx≤‖x‖‖y‖∗ for all x, which implies for all x, yTx−‖x‖≤0. Therefore x=0 is the value that maximizes yTx−‖x‖, with maximum value 0.
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