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What is the dual of a norm that is the sum of two-norms? Specifically, say we have the following norm for a $\mathbf{x}\in \mathbb{R}^n$ and $\mathbf{A}_i \in \mathbb{R}^{m \times n}$

$\|\mathbf{x}\| = \displaystyle{ \sum_{i=0}^{n} \|\mathbf{A}_i \cdot \mathbf{x} \|_2}$.

How would you then find

$\|\mathbf{y}\|_* = \underset{\mathbf{x}}{\mathrm{max}} \left\{ |\mathbf{y}^T \cdot \mathbf{x}| \;\; \mathrm{s.t.} \;\; \|\mathbf{x}\| \leq 1\right\}$?

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This doesn't define a norm. For example think of the case where $x \in \ker \begin{bmatrix}A_1\\.\\.\\.\\A_n\end{bmatrix}\setminus \{0\}$. – dohmatob Mar 16 at 14:03
    
I am facing a similar problem: having the norm $ \| u \|_{_H^1}} = \| u \|_{_{L^2}} + \| u' \|_{_{L^2}} $ what is then the norm $\| u \|_{_{H^1'}}$? – dktr.k1 Apr 18 at 7:40

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