<< 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.


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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

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?