Dual norm
For a given norm on , the dual norm, denoted , is the function from 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 ; it is homogeneous of degree , that is, for every and .
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 -norm is the -norm. This is because the inequality
holds trivially and is attained for .
- The dual norm above is the original norm we started with. (The proof of this general result is more involved.)