N dimensional complex space¶
In this section we review important features of N dimensional complex vector space \(\CC^N\).
An element \(x\) in \(\CC^N\) is written as
where each \(x_i\) is a complex number.
Vector space operations on \(\CC^N\) are defined by:
\(\CC^N\) comes with the standard ordered basis \(B = \{e_1, e_2, \dots, e_N\}\):
We note that the basis is same as the basis for \(N\) dimensional real vector space (the Euclidean space).
An arbitrary vector \(x\in\CC^N\) can be written as
Inner product¶
Standard inner product is defined as:
where \(\overline{y_i}\) denotes the complex conjugate.
This makes \(\CC^N\) an inner product space.
This satisfies the inner product rule:
Norm¶
The length of the vector (a.k.a. \(\ell_2\) norm) is defined as:
This makes \(\CC^N\) a normed linear space.
Distance¶
Distance between two vectors is defined as:
This makes \(\CC^N\) a metric space.
\(\ell_p\) norms¶
In addition to standard Euclidean norm, we define a family of norms indexed by \(p \in [1, \infty]\) known as \(\ell_p\) norms over \(\CC^N\).
\(\ell_p\) norm is defined as:
So we have:
\(\ell_1\) norm¶
From above definition we have
We use norms as a measure of strength of a signal or size of an error. Different norms signify different aspects of the signal.
Quasi-norms¶
In some cases it is useful to extend the notion of \(\ell_p\) norms to the case where \(0 < p < 1\).
In such cases norm as defined in (2) doesn’t satisfy triangle inequality, hence it is not a proper norm function. We call such functions as quasi-norms.
\(\ell_0\) “norm”¶
Of specific mention is \(\ell_0\) “norm”. It isn’t even a quasi-norm. Note the use of quotes around the word norm to distinguish \(\ell_0\) “norm” from usual norms.
\(\ell_0\) “norm” is defined as:
where \(\supp(x) = \{ i : x_i \neq 0\}\) denotes the support of \(x\).
Note that \(\| x \|_0\) defined above doesn’t follow the definition in (2).
Yet we can show that:
which justifies the notation.