Dual Norm

For a given norm |cdot| on $mathbf{R}^n$ , the dual norm, denoted |cdot|_ast , is the function from $mathbf{R}^n$ to with values

The above definition indeed corresponds to a norm: it is convex, as it is the pointwise maximum of convex (in fact, linear) functions x rightarrow y^Tx ; it is homogeneous of degree , that is, |alpha x|_ast = alpha |x|_ast for every $x in mathbf{R}^n$ and alpha ge 0 .

By definition of the dual norm,

This can be seen as a generalized version of the Cauchy-Schwartz inequality, which corresponds to the Euclidean norm.

Examples:

The norm dual to the Euclidean norm is itself. This comes directly from the Cauchy-Schwartz inequality.
The norm dual to the the -norm is the -norm. This is because the inequality

holds trivially, and is attained for x = mbox{bf sign}(y) .

The dual to the dual norm above is the original norm we started with. (The proof of this general result is more involved.)