<< Chapter < Page Chapter >> Page >
This module includes a brief introduction to metric spaces at a suitable level of detail for study of signals and systems.

Introduction

In may courses, concepts such as continuity and convergence are invoked without much discussion of their formal definitions, instead relying on the reader's intuitive understanding of these matters. However, for purposes of proofs, including some in the main body of material for this course, a greater rigor is required. This module will discuss metric spaces, a mathematical construct that provide a framework for the study continuity, convergence, and other related ideas in their most concrete but still formal senses. This is accomplished by formalizing a notion of the distance between two elements in a set. The intent in this and subsequent modules in this chapter is not to give a complete overview of the basic topics of analysis but to give a short introduction to those most important to discussion of signal processing in this course.

Metric spaces

A notion of distance

In many situations in signal processing it is often useful to have a concept of distance between the points in a set. This notion is mathematically formalized through the idea of a metric space. A metric space ( M , d ) is a set M together with a function d : M × M R that assigns distances between pairs of elements in M while satisfying three conditions. First, for every x , y M , d ( x , y ) 0 with d ( x , y ) = 0 if and only if x = y . Second, for every x , y M , d ( x , y ) = d ( y , x ) symmetrically. Third, for every x , y , z M , d ( x , y ) + d ( x , z ) d ( y , z ) , which is known as the triangle inequality.

There are, of course, several different possible choices of definitions for distances in a given set. Our typical intuitive understanding of distance in R n fits within this framework as the standard Euclidean metric

d ( x , y ) = | | x - y | | 2

as does the taxicab or Manhatten metric

d ( x , y ) = | | x - y | | 1

that sums individual components of vectors, representing, for example, distances traveled walking around city blocks. Another simple yet more exotic example is provided by the discrete metric on any set defined by

d ( x , y ) = 0 x = y 1 x y

in which all pairs of distinct points are equidistant from eachother but every point is distance zero from itself. One can check that these satisfy the conditions for metric spaces.

Relationship with norms

It is not surprising that norms, which provide a notion of size, and metrics, which provide a notion of distance, would have a close relationship. Intuitively, one way of defining the distance between two points in a metric space could be the size of their difference. In other words given a vector space V over the field F with norm | | · | | , we might ask if the function

d ( x , y ) = | | x - y | |

for every x , y V satisfies the conditions for ( V , d ) to be a metric space.

Let V be a vector space over the field F with norm | | · | | , and let d ( x , y ) = | | x - y | | . Recall that since | | · | | is a norm, | | x | | = 0 if and only if x = 0 and | | a x | | = | a | | | x | | for all a F and x V . Hence | | x - y | | 0 for all x , y V and | | x - y | | = 0 if and only if x = y . Since y - x = - ( x - y ) and | | - ( x - y ) | | = | | x - y | | it follows that | | x - y | | = | | y - x | | for all x , y V . Finally, | | x | | + | | y | | | | x + y | | by the properties of norms, so | | x - y | | + | | x - z | | | | y - z | | for all x , y , z V . Thus, ( V , d ) does indeed satisfy the conditions to be a metric space and is discussed as the metric space induced by the norm | | · | | .

Metric spaces summary

Metric spaces provide a notion of distance and a framework with which to formally study mathematical concepts such as continuity and convergence, and other related ideas. Many metrics can be chosen for a given set, and our most common notions of distance satisfy the conditions to be a metric. Any norm on a vector space induces a metric on that vector space and it is in these types of metric spaces that we are often most interested for study of signals and systems.

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

Ask