<< Chapter < Page Chapter >> Page >

While vector spaces have additional structure compared to a metric space, a general vector space has no notion of “length” or “distance.”

Definition 1

Let V be a vector space over K . A norm is a function · : V R such that

  • x 0 x V
  • x = 0 iff x = 0
  • α x = | α | x x V , α K
  • x + y x + y x , y V

A vector space together with a norm is called a normed vector space (or normed linear space ).

  • V = R N : x 2 = i = 1 N | x i | 2
    An illustration showing a point x in R2 and it's ell_2 (Euclidian) norm.  The norm is equal to the length of a straight line connecting x to the origin.
  • V = R N : x 1 = i = 1 N | x i | (“Taxicab”/“Manhattan” norm)
  • V = R N : x = max i = 1 , . . . , N | x i |
    An illustration showing a point x in R2 and it's ell_infinity norm.  The norm is equal to the length of the longer of the two (orthogonal) paths that connect x to the x- and y-axes.
  • V = L p [ a , b ] , p [ 1 , ) : x ( t ) p = a b | x ( t ) | p d t 1 / p (The notation L p [ a , b ] denotes the set of all functions defined on the interval [ a , b ] such that this norm exists, i.e., x ( t ) p < .)

Note that any normed vector space is a metric space with induced metric d ( x , y ) = x - y . (This follows since x - y = x - z + z - y x - z + y - z .) While a normed vector space “feels like” a metric space, it is important to remember that it actually satisfies a great deal of additional structure.

Technical Note: In a normed vector space we must have (from N2) that x = y if x - y = 0 . This can lead to a curious phenomenon when dealing with continuous-time functions. For example, in L 2 ( [ a , b ] ) , we can consider a pair of functions like x ( t ) and y ( t ) illustrated below. These functions differ only at a single point, and thus x ( t ) - y ( t ) 2 = 0 (since a single point cannot contribute anything to the value of the integral.) Thus, in order for our norm to be consistent with the axioms of a norm, we must say that x = y whenever x ( t ) and y ( t ) differ only on a set of measure zero. To reiterate x = y x ( t ) = y ( t ) t [ a , b ] , i.e., when we treat functions as vectors, we will not interpret x = y as pointwise equality, but rather as equality almost everywhere .

A smooth function defined on the interval [-1,1]. A function that is identical to the previous function, except for a point discontinuity where it takes a different value.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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 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, Digital signal processing. OpenStax CNX. Dec 16, 2011 Download for free at http://cnx.org/content/col11172/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing' conversation and receive update notifications?