Dual NormFor 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:
holds trivially, and is attained for .
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