7. Norms#
Definition#
Let’s look at the concept of norms and normed vector spaces.
Definition
A norm is a special map between vector spaces which takes element from a vector space \((V, \mathbb{F})\) and maps those elements to the positive real line \(\mathbb{R}_{+}\)
We call a normed vector space is a vector space which has a norm defined on it.
Definition
A normed vector space is a vector space which has a norm defined on it.
Properties of norm#
A norm \(||\cdot||\) that maps \(V \rightarrow \mathbb{R}_+\) satisfies the following properties:
Triangle property: $\( || v_1 + v_2 || \le ||v_1|| + ||v_2|| \forall v_1, v_2 \in V \)$
\(||\alpha v|| = |\alpha| ||v||\)
Note
As the normed vector space \(V\) has the associated field \(\mathbb{F}\), the value of \(|\alpha|\) depends on \(\mathbb{F}\). If \(\mathbb{F}\) is real number, then \(|\alpha|\) is absolute value. Else if \(\mathbb{F}\) is field of complex number (\(\mathbb{C}\)) then \(|\alpha|\) will be the magnitude of the complex number.
\(||v|| = 0 \leftrightarrow v = \theta_v\)
Remark
A norm has to satisfy the above properties. Thus to check if a candidate norm is in fact a norm, we can check to see if it satifies all the above properties.
Norms for vector space#
Example: Given the normed vector space \((\mathbb{F}^n, \mathbb{F})\)
The 1-norm: \(||x||_1 = \Sigma_{i=1}^{n} |x_i|\)
The 2-norm: \(||x||_2 = (\Sigma_{i=1}^{n} |x_i|^2)^{\frac{1}{2}}\)
The p-norm: \(||x||_p = (\Sigma_{i=1}^{n} |x_i|^p)^{\frac{1}{p}}\)
The \(\infty\)-norm: \(||x||_\infty = \max_{i} |x_i|\)
These are the norms that we use frequently in \(\mathbb{R}^n\).
How about vector space with matrices? We can construct norms from those matrices:
Norms for vector space with matrices#
Example: Consider matrix \(A \in \mathbb{F}^{m \times n}\)
The a-norm: \(||A||_a = \Sigma_{i=1}^{m} \Sigma_{j=1}^{n} |a_{ij}|\)
The Frobenius norm: \(||A||_F = (\Sigma_{i=1}^{m} \Sigma_{j=1}^{n} |a_{ij}|^2)^{\frac{1}{2}}\)
The b-norm: \(||A||_b = \max_{i \in \{1 \dots m\}, j \in \{1 \dots n\}} |a_{ij}|\)
Though we do not use these norms so much. In linear systems, we tend to use another definition of norm called the induced norm, which will be discussed later.
What about function space? Can we have norms for function space?
Norms for function space#
Example: Given functions \(f(\cdot) \in C([t_0, t_1], \mathbb{F}^n)\) (\(C\) stands for continuous function). Possible norms are:
The 1-norm (the \(L_1\) norm for functions): \(||f||_1 = \int_{t_0}^{t_1} ||f(t)|| dt\).
Here interestingly, the norm \(||f_1||\) inside the integral does not have to be the 1-norm, but it can rather be any of the norm that we have previously discussed.
The 2-norm (the \(L_2\) norm for functions): \(||f||_2 = (\int_{t_0}^{t_1} ||f(t)||^2 dt) ^ \frac{1}{2}\)
The \(\infty\)-norm: \(||f||_\infty = \max \{||f(t)||, t \in [t_0, t_1]\}\)
Relations between norms#
If we have a vector that is finite in 1 norm, do we know anything about this vector in another norm?
Remark: Equivalence of norms
Two norms \(||\cdot||_a, ||\cdot||_b\) on \((V, \mathbb{F})\) are said to be equivalent if one can be bounded with respect to another. We write this as \(\exists m_l, m_u \in \mathbb{R}_+\) s.t
From here, we have some nice relationships
Example: Given a vector \(x \in \mathbb{R}^n\):