Highly Nonlinear Approximations for Sparse Signal Representation

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Subspaces - Direct sum

A subset $ {\cal{S}}$ of a vector space $ {\cal{V}}$ is a subspace of $ {\cal{V}}$ if it is a vector space with respect to the vector space operations on $ {\cal{V}}$. A subspace which is a proper subset of the whole space is called a proper subspace. Two subspaces $ {\cal{V}}_1$ and $ {\cal{V}}_2$ are complementary or disjoint if $ {\cal{V}}_1 \cap {\cal{V}}_2 = \{0\}$.

The sum of two subspaces $ {\cal{V}}_1$ and $ {\cal{V}}_2$ is the subspace $ {\cal{V}}= {\cal{V}}_1 + {\cal{V}}_2$ of elements $ v=v_1 + v_2, v_1 \in {\cal{V}}_1, v_2\in {\cal{V}}_2$. If the subspaces $ {\cal{V}}_1$ and $ {\cal{V}}_2$ are complementary $ {\cal{V}}= {\cal{V}}_1 + {\cal{V}}_2$ is called direct sum and indicated as $ {\cal{V}}= {\cal{V}}_1 \oplus {\cal{V}}_2$. This implies that each element $ v \in {\cal{V}}$ has a unique decomposition $ v=v_1 + v_2, v_1 \in {\cal{V}}_1, v_2\in {\cal{V}}_2$.

For the sets $ V_1$ and $ V_2$, the set $ \{v \in V_1 : v \not\in V_2\}$ is denoted by $ V_1 \setminus V_2$.