Highly Nonlinear Approximations for Sparse Signal Representation

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Normed vector space

A norm $ \Vert\cdot\Vert$ on a vector space $ {\cal{V}}$ is a function from $ {\cal{V}}$ to $ \mathbb{R}$ such that for every $ v,w\in {\cal{V}}$ and $ a \in \mathbb{F}$ the following three properties are fulfilled

  1. $ \Vert v \Vert \ge 0,$ and $ \Vert v \Vert=0 \Longleftrightarrow v=0$.
  2. $ \Vert a v \Vert= \vert a\vert \Vert v \Vert.$
  3. $ \Vert v + w \Vert \le \Vert v \Vert + \Vert w \Vert.$
A vector space $ {\cal{V}}$ together with a norm is called a normed vector space.