Pre-req - [Norm, for Beginners](../../AMATH%20584%20Numerical%20Linear%20Algebra/Matrix%20Theory/Norm,%20for%20Beginners.md). --- ### **Intro** Dual norm denoted as $\Vert \cdot\Vert_\star$ is defined for a norm $\Vert \cdot\Vert$ in $\mathbb E$. Here it's the definition: $ \Vert y\Vert_\star := \max_{x}\left\lbrace \langle x, y\rangle: \Vert x\Vert \le 1 \right\rbrace $ Which is just simply, the objective value of a linear program on the convex set $\Vert x\Vert$ defined by the norm ball in the Euclidean space. Note, it's also the support function of a specific unit norm ball. See [[Support Function]] for more info. The idea is very useful for duality and looking for the prox of some functions that involves the use of some type of norms. --- ### **Generalized Cauchy-Schwarz Inequality** **Theorem Statement**: $ \begin{aligned} |\langle x, y\rangle| \le \Vert x\Vert\Vert y\Vert_\star \end{aligned} $ The proof is direct from the definition of dual norm: $ \begin{aligned} \Vert y\Vert_\star &\ge \langle y, x\rangle\; \forall x: \Vert x\Vert \le 1 \\ \implies \Vert y\Vert_\star &\ge \frac{\langle y, x\rangle}{\Vert x\Vert} \; \forall x \neq \mathbf 0 \end{aligned} $ The absolute value can be obtained by considering $\Vert - y\Vert_\star$. $\Vert -y\Vert_\star = -\min_x\{\langle x, y\rangle: \Vert x \Vert \le 1\}$. --- ### **P-Norm Dual** The dual of the p-norm $\Vert y\Vert_p^\star$, is $\Vert y\Vert_{q}$ such that $1/p + 1/q = 1$. In the special case we have $(\Vert y\Vert_\infty)^\star = \Vert y\Vert_1$ **Proof** An exercise for the reader, which is likely to be myself. --- ### **Dual of The Induced Energy Norm** **#UNFINISHED: Look for the Dual of the induced energy norm**. Source: \<First Order method in Optimizations\> By SIAM Example 1.6, they a quick example. --- ### **Misc** To understand dual norm in a more abstract context, i.e: As the dual space of formed by linear functional of a Banach space, visit [Linear Functionals and Dual Spaces](../../MATH%20601%20Functional%20Analysis,%20Measure%20Theory/Linear%20Functionals%20and%20Dual%20Spaces.md).