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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

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

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

$\displaystyle \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

$\displaystyle \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

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

of real numbers satisfying the convergence condition

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

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

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

One can show that (Minkowski's inequality)

$\displaystyle \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,

$\displaystyle \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

$\displaystyle \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

$\displaystyle \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

$\displaystyle 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$ .
\begin{figure}\centering\epsfig{file=pythagorean_circle.eps,scale=.5}\end{figure}

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

$\displaystyle \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

$\displaystyle 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$ .
\begin{figure}\centering\epsfig{file=hamming_circle.eps,scale=.5}\end{figure}

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

$\displaystyle \lim_{p\to\infty}
\left(\sum^n_{k=1}\vert x_k\vert^p\right)^{\frac{1}{p}}$ $\displaystyle =$ $\displaystyle 1
\quad\Longrightarrow$  
$\displaystyle \textrm{Max}\{\vert x_1\vert,\vert x_2\vert,\cdots ,\vert x_n\vert\}$ $\displaystyle =$ $\displaystyle 1\quad
\textrm{(unit sphere in }R^n\textrm{ with }\ell^\infty\textrm{-norm)}~.$  

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

$\displaystyle d(x,y)$ $\displaystyle =\lim_{p\to\infty} \left(\sum^n_{k=1}\vert x_k-y_k\vert^p\right)^{\frac{1}{p}}$    
  $\displaystyle =\textrm{Max}\{\vert x_1-y_1\vert,\vert x_2-y_2\vert,\cdots ,\vert x_n-y_n\vert\}~.$    

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 $ .
\begin{figure}\centering\epsfig{file=chebyshev_circle.eps,scale=.5}\end{figure}


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Next: Metric Spaces Up: Infinite Dimensional Vector Spaces Previous: Inner Product Spaces   Contents   Index
Ulrich Gerlach 2010-12-09