Highly Nonlinear Approximations for Sparse Signal Representation

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Inner product space

An inner product on a vector space $ {\cal{V}}$ is a map from $ {\cal{V}}$ to $ \mathbb{F}$ which satisfies the following axioms

  1. $ \langle v , v \rangle \ge 0,\; v\in {\cal{V}}$, and $ \langle v , v \rangle =0 \Longleftrightarrow v =0.$
  2. $ \langle v + w , z \rangle = \langle v , z \rangle + \langle w , z \rangle ,\; \; v,w,z\in {\cal{V}}.$
  3. $ \langle v , a z \rangle = a \langle v , z \rangle , \; v,z\in {\cal{V}}$ and $ a \in \mathbb{F}$.
  4. $ \langle v , w \rangle = \overline { \langle w , v \rangle }, \; v,w\in {\cal{V}}$.
A vector space $ {\cal{V}}$ together with an inner product $ \langle \cdot , \cdot \rangle $ is called an inner product space.