"

Dual norm

For a given norm ||\cdot|| on \mathbb{R}^n, the dual norm, denoted ||\cdot||_*, is the function from \mathbb{R}^n to \mathbb{R} with values

    \begin{align*} \|y\|_* &= \max\limits_{x}x^Ty: \|x\| \leq 1. \end{align*}

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 1, that is, ||\alpha x||_* = \alpha ||x||_* for every x \in \mathbb{R}^n and \alpha \ge 0.

By definition of the dual norm,

    \begin{align*} x^Ty &\leq \|x\|\cdot \|y\|_*. \end{align*}

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 l_\infty-norm is the l_1-norm. This is because the inequality

    \begin{align*} x^Ty &\leq \|x\|_\infty \cdot \|y\|_1. \end{align*}

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

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

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Linear Algebra and Applications Copyright © 2023 by VinUiversity is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Share This Book