Normed Linear Spaces

There exist other structures which a vector space may have. A norm on the vector space ${\cal V}$ is a linear functional, say $p(f)$ , with the following three properties:

1. Positive definiteness:  $p(f)>0$ for all nonzero vectors $f$ in ${\cal V}$ , and $p(f)=0\Leftrightarrow f=\vec 0$ .

2. Linearity:   $p(\alpha f)=\vert\alpha\vert p(f)$ for all vectors $f$ and for all complex numbers $\alpha$ .

3. Triangle inequality:   $p(f+g)\le p(f)+p(g)$ for all vectors $f$ and $g$ in ${\cal V}$ .

Such a function is usually designated by $p(f)=\Vert f\Vert$ , a norm of the vector $f$ . The existence of such a norm gives rise to the following definition:

A linear space ${\cal V}$ equipped with a norm $p(f)=\Vert f\Vert$ is called a normed linear space.

Example 1: Every inner product of an inner product space determines the norm given by

$$\Vert f\Vert =(\langle f,f\rangle )^{\frac{1}{2}} \quad ,$$

which, as we have seen, satisfies the triangle inequality,

$$\Vert f+g\Vert\le\Vert f\Vert +\Vert g\Vert\,.$$

Thus an inner product space is always a normed linear space with the inner product norm. However, a normed linear space is not necessarily an inner product space.

Lecture 3

Example 2: Consider the vector space of $n\times n$ matrices $A=[a_{ij}]$ . Then

$$\Vert A\Vert =\max_{i,j}\vert a_{ij}\vert$$

is a norm on this vector space.

Example 3: Consider the vector space of all infinite sequences

$$x=(x_1,x_2,\dots ,x_k,\dots )$$

of real numbers satisfying the convergence condition

$$\sum^\infty_{k=1}\vert x_k\vert^p<\infty$$

where $p\ge 1$ is a real number. Let the norm be defined by

$$\Vert x\Vert =\left(\sum^\infty_{k=1}\vert x_k\vert^p\right)^{\frac{1}{p}}~~.$$

One can show that (Minkowski's inequality)

$$\left(\sum^\infty_{k=1}\vert x_k+y_k\vert^p\right)^{\frac{1}{p}}\... ...\frac{1}{p}}+\left(\sum^\infty_{k =1}\vert y_k\vert^p\right)^{\frac {1}{p}}\,,$$

i.e., that the triangle inequality,

$$\Vert x+y\Vert\le \Vert x\Vert +\Vert y\Vert\,,$$

holds. Hence $\Vert~\cdot~\Vert$ is a norm for this vector space. The space of $p$ -summable $\left(\sum\limits^\infty_1 \vert x_k\vert^p<\infty \right)$ real sequences equipped with the above norm is called $\ell^p$ and the norm is called the $\ell^p$ -norm.

This $\ell^p$ -norm gives rise to geometrical objects with unusual properties. consider the following

Example 4: The surface of a unit sphere centered around the origin of a linear space with the $\ell^p$ -norm is the locus of points $\{ (x_1,x_2,\cdots\}$ for which

$$\left(\sum^\infty_{k=1}\vert x_k\vert^p\right)^{\frac{1}{p}}=1~.$$

Consider the intersection of this sphere with the finite dimensional subspace $R^n$ , which is spanned by $\{ (x_1,x_2,\cdots,x_n\}$ .

a) When $p=2$ , this intersection is the locus of points for which

$$\vert x_1\vert^2+\vert x_2\vert^2+\cdots +\vert x_n\vert^2=1\quad \textrm{(unit sphere in }R^n\textrm{ with }\ell^2\textrm{-norm)}$$

This is the familiar ($n-1$ )-dimensional unit sphere in $n$ -dimensional Euclidean space whose distance function is the Pythagorean distance

$$d(x,y)=\left\{\vert x_1-y_1\vert^2+\vert x_2-y_2\vert^2+\cdots +\vert x_n-y_n\vert^2\right\}^\frac{1}{2}~.$$

Figure 1.1: Circle in $R^2$ endowed with the Pythagorean distance function of $\ell ^2$ .

b) When $p=1$ , this intersection is the locus of points for which

$$\vert x_1\vert+\vert x_2\vert+\cdots +\vert x_n\vert=1\quad \textrm{(unit sphere in }R^n\textrm{ with }\ell^1\textrm{-norm)}$$

This is the ($n-1$ )-dimensional unit sphere in $n$ -dimensional vector space endowed with a different distance function, namely one which is the sum of the differences

$$d(x,y)=\vert x_1-y_1\vert+\vert x_2-y_2\vert+\cdots +\vert x_n-y_n\vert~,$$

between the two locations in $R^n$ , instead of the sum of squares. This distance function is called the Hamming distance, and one must use it, for example, when travelling in a city with a rectangular grid of streets. With such a distance function a circle in $R^2$ is a square standing on one of its vertices. See Figure 1.2. A 2-sphere in $R^3$ is a cube standing on one of its vertices, etc.

Figure 1.2: Circle in $R^2$ endowed with the Hamming distance function of $\ell ^1$ .

c) When $p\to\infty$ , this intersection is the locus of points for which

$$\lim_{p\to\infty} \left(\sum^n_{k=1}\vert x_k\vert^p\right)^{\frac{1}{p}} = 1 \quad\Longrightarrow$$

$$\textrm{Max}\{\vert x_1\vert,\vert x_2\vert,\cdots ,\vert x_n\vert\} = 1\quad \textrm{(unit sphere in }R^n\textrm{ with }\ell^\infty\textrm{-norm)}~.$$

Such a unit sphere $R^n$ is based on the distance function

$$d(x,y) =\lim_{p\to\infty} \left(\sum^n_{k=1}\vert x_k-y_k\vert^p\right)^{\frac{1}{p}}$$

$$=\textrm{Max}\{\vert x_1-y_1\vert,\vert x_2-y_2\vert,\cdots ,\vert x_n-y_n\vert\}~.$$

This is called the Chebyshev distance. It is simply the maximum coordinate difference regardless of any other differences. With such a distance function a circle in $R^2$ is a square. See Figure 1.3. A 2-sphere in $R^3$ is a cube, etc.

Figure 1.3: Circle in $R^2$ endowed with the Chebyshev distance function of $\ell ^\infty$ .