| # 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 the -norm is the -norm. This is because the inequality holds trivially, and is attained for . The dual to the dual norm above is the original norm we started with. (The proof of this general result is more involved.)