# Highly Nonlinear Approximations for Sparse Signal Representation

## Subspaces - Direct sum

A subset of a vector space is a **subspace**
of if it is a vector space with respect to
the vector space operations on . A subspace which is a
proper subset of the whole space is called a
**proper subspace**.
Two subspaces
and
are **complementary** or **disjoint**
if
.

The sum of two subspaces
and
is the
subspace
of elements
. If the
subspaces
and
are
*complementary*
is called
**direct sum** and indicated as
. This implies
that each element
has
a unique decomposition
.

For the sets and , the set is denoted by .