Gramian matrix: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
 
Line 1: Line 1:
{{About|linear algebra and analysis|field theory|Field norm|ideals|Ideal norm|group theory|Norm (group)|norms in descriptive set theory|prewellordering}}
== Nike Free Run  for that religious bunch ==


In [[linear algebra]], [[functional analysis]] and related areas of [[mathematics]], a '''norm''' is a [[function (mathematics)|function]] that assigns a strictly positive ''length'' or ''size'' to each [[vector (mathematics and physics)|vector]] in a [[vector space]], other than the [[zero vector]] (which has zero length assigned to it). A '''seminorm''', on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector).
Besides that, it's about 15 megabytes. I'm the only other person that has it, besides Carlos. I recieve it. Nonetheless the conflict and dysfunction among the crew of SGU was still true to life and closer to adulthood than te Stargate franchise will ever get. <br><br>Making a new record each time a coverage change occurs recording the start and ending dates and times of each plan storing the historical records in a history table unioning the historical records and also the current records together to obtain a full report Figure 2 shows a business relationship diagram (ERD) that illustrates my suggestions. The diagram is really a basic model of the solution; to prevent confusion, I've minimized the attribute set. <br><br>I always look forward to reading your newsletters and e-mail to see what new project your chapter volunteers are planning. Although I live 2 hours away, I have made several trips for your meetings for your excellent presentations. You can also indicate the person you are looking for here if you want to find their postcode. Just type the name of the person. <br><br>Seven things I learned at IITP 2013. The summary of this year's Institute of IT [http://www.uluruexpress.com.au/flux-system/plugins/formfile.asp Nike Free Run] Professionals 2013 Conference was first published last weekend and was popular enough to entitled to the previous week's top stories as well as heading this week's list. <br><br>Lets get it done this way: If all the species of the planet relied solely on homosexual acts the world would have no life, who have been better for the planet. Or, for the religious bunch: If God (when there [http://www.bellaflora.com.au/settings/maintenance.asp Abercrombie Australia] is a God) created Adam and Eve how come they are of different genders? Answer: God created Adam and Eve for reproduction to be able to procreate the planet with unhappy people. <br><br>Planning Status R Garage courts are in good condition. Retain parking use Garage courts are in good condition. Next Mass Effect Serves as Sequel to Main Trilogy, New Races Get Details  Report Download UC Browser 9.3 for Java Download BBM for Android 1.0.2.83 Who Needs Winamp When you are able Play Music from the Terminal Download UC Browser 9.2 for Java 14 Alleged Members of Anonymous and RedHack Referred to Ankara Court Top Five Microsoft Funny Quotes About [http://www.uluruexpress.com.au/flux/upgrade/content.asp Oakley Holbrook] Linux Samsung Galaxy Grand 2 Goes Official with DualSIM Support and 5.25Inch Display Windows 9 Concepts: Designers Think of [http://www.smallbusinesssolutions.qld.gov.au/common/functions.asp Nike Air Max 2013] the Next Microsoft Operating System Microsoft Confirms Windows 9 Could Merge Modern OS VersionsScribd, the popular document sharing site has announced a new round of funding, raising $13 million from a number of institutional investors. This is the company third funding round and it plans to use the money to fund growth, by hiring several new people, mainly in the engineering department.Scribd enables users to upload and share documents with other users or make them completely public, the same as Flickr or YouTube for documents.MLC Investments and SVB Capital led the round with existing investors Redpoint Ventures, Charles River Ventures and Kinsey Hills Group also participating.It last funding round closed a couple of years ago, it raised $9 million in December 2008.<ul>
 
  <li>[http://blacktr.co.vu/bbs/forum.php?mod=viewthread&tid=721627 http://blacktr.co.vu/bbs/forum.php?mod=viewthread&tid=721627]</li>
 
  <li>[http://shanafanghua.imotor.com/viewthread.php?tid=37020&extra= http://shanafanghua.imotor.com/viewthread.php?tid=37020&extra=]</li>
 
  <li>[http://chandipi.net/thread-52477-1-1.html http://chandipi.net/thread-52477-1-1.html]</li>
 
  <li>[http://www.wikisportsmemorabilia.com/User:Esevpdwp#Nike_Australia_Please_view_this_online_seminar http://www.wikisportsmemorabilia.com/User:Esevpdwp#Nike_Australia_Please_view_this_online_seminar]</li>
 
  <li>[http://radoslawsobik.pl/rybnik360/spip.php?article1&archives=2013-05-01/ http://radoslawsobik.pl/rybnik360/spip.php?article1&archives=2013-05-01/]</li>
 
</ul>


A simple example is the 2-dimensional [[Euclidean space]] '''R'''<sup>2</sup> equipped with the [[#Euclidean norm|Euclidean norm]]. Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional [[cartesian coordinate system]] starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the [[magnitude (mathematics)|magnitude]].
== Oakley Sunglasses Cheap ==


A vector space with a norm is called a [[normed vector space]]. Similarly, a vector space with a seminorm is called a seminormed vector space.
Products for Geeks, Gamers, and General Users The brand new site is segmented into eight distinct product categories for all sorts of PC and Mac users: Memory: the easiest to install and most costeffective performance upgrade, adding more memory can dramatically [http://www.bellaflora.com.au/common/update.asp Oakley Sunglasses Cheap] improve app and system responsiveness, gaming, and specific memory intensive usages for example video streaming over wireless networks. Graphics Cards: gamers looking for the ultimate experience can choose from a variety [http://www.romagnolabeefgenetics.com.au/common/standard.asp Nike Air Max 90] of options including the R9 290X, the world's fastest graphics card. <br><br>Since his immunity system starts to dysfunction he would need extra. If you want to know about the nutrition factor of the AIDS patient, visit the site. A little vending machine business could be started on quite a low budget therefore it is better to start out slowly and face a smaller amount of risk. After that you can choose to invest further to some great extent as soon as the business starts to earn a certain amount of income, and after you determine that vending is really the business that you desire to do.. <br><br>Deuces Cracked have place a lot of effort into developing a well stocked catalog of full ring quite happy with over 120 videos already available. Furthermore, new content is produced on a weekly basis by full ring specialists such as newmanmi, bottomset, BalugaWhale, goldseraph, RapidEvolution and DjSensei. <br><br>Mucha has been hired as Managing Editor in charge of correspondents. Mucha will oversee GlobalPost' corps of correspondents, multimedia, and can contribute to planning and executing special features and reporting projects. "I stopped taking them. A part of that created in me this nervous anxiety, and that is when I was really screwed. <br><br>Not to mention that its not that many guys. Have a small break to cool down after which check out the steam room (parilka), where one can be beaten about the body with dried branches (vennik), usually birch, oak or juniper. Thirteen years later the S 500 has still not exceeded that 2000 high by a decisive margin. For this type of breakout to occur, it would need to break not only the 2000 high but also the marginally higher record of 2007, by a number of percentage points. <br><br>I know that [http://www.completionpress.com.au/graphics/popups/frameset.asp Air Max Thea Australia] in primary he didn't like to say anything or ask anything just in case he was wrong or got laughed at (bullying issue) but i'd really love it if he felt [http://www.romagnolabeefgenetics.com.au/common/standard.asp Nike Air Max Australia] able to be himself more. He is clever, funny and generally a nice lad.. Some minisites converts constantly while others get only occasional sales. What makes such a huge difference? It is something related to design layout.<ul>
 
  <li>[http://www.mingjiuxuan.com/news/html/?103476.html http://www.mingjiuxuan.com/news/html/?103476.html]</li>
 
  <li>[http://www.wjiaq.com/news/html/?374657.html http://www.wjiaq.com/news/html/?374657.html]</li>
 
  <li>[http://spammer-fangen.de/phpbb-201406/viewtopic.php?f=2&t=182032 http://spammer-fangen.de/phpbb-201406/viewtopic.php?f=2&t=182032]</li>
 
  <li>[http://bbs.yangshuo.net/forum.php/forum.php?mod=viewthread&tid=336162 http://bbs.yangshuo.net/forum.php/forum.php?mod=viewthread&tid=336162]</li>
 
  <li>[http://www.wikimusicmemorabilia.com/User:Vwcdjsrah#Nike_Air_Jordan_Australia http://www.wikimusicmemorabilia.com/User:Vwcdjsrah#Nike_Air_Jordan_Australia]</li>
 
</ul>


==Definition==
== Beats Headphones Australia  huts ==
Given a [[vector space]] ''V'' over a [[field (mathematics)|subfield]] ''F'' of the [[complex number]]s, a '''norm on''' ''V'' is a [[function (mathematics)|function]] {{nowrap|''p'': ''V'' → '''R'''}} with the following properties:<ref>{{harvnb|Prugovečki|1981|loc=[http://books.google.com/books?id=GxmQxn2PF3IC&pg=PA20 page 20]}}</ref>


For all ''a'' ∈ ''F'' and all '''u''', '''v''' ∈ ''V'',
Please configure your browser or system settings to permit popups on CSUMentor. Federal Student Aid [http://www.sitzler.com.au/settings/display.asp Beats Headphones Australia] provides publications, online tools, [http://www.nadya.com.au/flux/core/styles/library.asp Ray Ban Sunglasses Australia] videos, and other resources to help you prepare and purchase college or career school. Forces are getting stronger. And they are getting ready for more, he said. <br><br>For one thing, he wading into volatile territory having a story set in the Middle East. You on really tricky political ground, and also you know no matter what you do, people have strongly held beliefs which will lead them to see the material through one lens or another, you just have to try to adhere really rigidly to the facts, he says.. <br><br>""Ye are the light around the globe. A city that is set on an hill cannot be hid. Topic galleries provide easy access to stories and photos about people, places, organizations, events and subjects of great interest to you. They bring together rich.. No training required. Lots of opportunities. <br><br>I don think that they have much to gain beyond having a deterrent [against any attack]. But the fear is that they will export nuclearweapons materials or technology, therefore we can ignore them either. If this sounds like too much there's an online design service (pounds 40) in which you send in detailed plans of the garden, what you want, and how much you really can afford. Expert Nicholas Roeber will tell you what to do. <br><br>Trustcash's President and Chief Executive Officer, Kent Carasquero, stated, "Using the best technology to support our new website must go hand in hand with bringing the Trustcash brand alive by infusing it with a provocative voice that is both distinct and compelling. Our goal would be to create a website that hits the mark for both the online and debit card clients. <br><br>At first Violet whistled him, correctly, for offensive interference and negated the basket. Then she changed her mind when the Knicks argued that Nate did not touch the ball (so it appeared he did not).. You mustn't misrepresent your relationship with St. Jude or present false or misleading impressions about St. <br><br>By the [http://www.gregoryburgessarchitects.com.au/Mobile/MasterPages/header.asp Air Max Australia] way, Im all for biting the hand [http://www.smallbusinesssolutions.qld.gov.au/common/functions.asp Nike Air Max 2014] that fed me. Angelfire blows. Roll it over the area you wish to cover and stake the edges using landscape stakes, that also is found at garden centers. Heavy landscape fabric also is effective when thrown over a compost heap. <br><br>Your main income and resources must come from what the game provides (trade specials, good starting locations, huts, etc.) and from interaction with the AI, by trading techs, and becoming tribute and/or gifts. You can't possibly do it all on your own..<ul>
# ''p''(''a'''''v''') = |''a''| ''p''('''v'''), (''[[Homogeneous_function|absolute homogeneity]]'' or ''absolute scalability'').
 
# ''p''('''u''' + '''v''') ≤ ''p''('''u''') + ''p''('''v''') (''[[triangle inequality]]'' or ''[[subadditive function|subadditivity]]'').
  <li>[http://elec.newroomschat.com/spip.php?article30/ http://elec.newroomschat.com/spip.php?article30/]</li>
# If ''p''('''v''') = 0 then '''v''' is the [[zero vector]] (''separates points'').
 
  <li>[http://arcib.dowling.edu/wiki/index.php/User:Dgsucsvo#Nike_Free_5.0_Womens__Anastasia http://arcib.dowling.edu/wiki/index.php/User:Dgsucsvo#Nike_Free_5.0_Womens__Anastasia]</li>
 
  <li>[http://www.ewmsj.com/news/html/?227987.html http://www.ewmsj.com/news/html/?227987.html]</li>
 
  <li>[http://f1fanclub.com.au/activity/p/26596/ http://f1fanclub.com.au/activity/p/26596/]</li>
 
  <li>[http://www.imc-e.net/thread-1396900-1-1.html http://www.imc-e.net/thread-1396900-1-1.html]</li>
 
</ul>


By the first axiom, absolute homogeneity, we have ''p''('''0''') = 0 and ''p''('''-v''') = ''p''('''v'''), so that by the triangle inequality
== Ralph Lauren Australia  who's in police custody ==
: ''p''('''v''') ≥ 0 (''positivity'').


A '''seminorm''' is a '''norm''' with the 3rd property (separating points) removed.
Although the serial [http://www.smallbusinesssolutions.qld.gov.au/custom/revision.asp Ralph Lauren Australia] number isn't externally listed, but it is available when plugged into a computer on the "Summary" tab in iTunes. If the display works enough to use the screen, the serial number also can be viewed by browsing Settings > General > About from inside the iOS. <br><br>Once you succeed locally, you succeed nationally,' says Gupta. Remember Aaj Tak? It began as a beacon of hope for small , local advertisers. In spite of its more undesired flavor, robusta beans are confirmed to be an organic pesticide and stimulant, also growing in places that the Arabica beans can not grow. Therefore, it's used as an inexpensive substitute for Arabica in commercial coffee blends and in nearly every instant coffee brand.. <br><br>U can also, with practice. Help p lvl someone. Having the bible text and notes in separate frames so the bible text can be read without distraction.6. The alignment from the bible text and the notes. [http://www.uluruexpress.com.au/flux/upgrade/content.asp Oakley Holbrook] Without these any machine is just a piece of metal or plastic. There are different types of software available for different uses. <br><br>This press release may be deemed to contain forwardlooking information. Any forwardlooking statements are made pursuant to the safe harbor provisions from the Private Securities Litigation Reform Act The non-public Securities Litigation Reform Act of 1995 (PSLRA) implemented several significant [http://www.nadya.com.au/graphics/popups/settings.asp Mont Blanc Australia] substantive changes affecting some instances brought under the federal securities laws, including changes associated with pleading, discovery, liability, class representation and awards fees as well as 1995. <br><br>It is absorbed through the skin and into the bloodstream. It can benefit your body keep estrogen and progesterone levels in balance, leading to maintained sense of equilibrium.. Her neck had bruise marks suggesting that attempts were made to strangle her."The rape accused Manoj, who is in police custody, had also raped his wife before marriage. The village panchayat had then ordered Manoj to marry the lady. <br><br>In my own experience, people can read as much [http://www.sitzler.com.au/settings/display.asp Beats Headphones Australia] as they want about something in advance, but it'll all venture out the window as soon as they do it for real. They'll still make a load of newbie mistakes once they try applying the skills they read about. <br><br>This is the basis of noise canceling, enabling you to escape your surroundings and listen to what you want. Sometimes however, this comes confined sticker price. Generally people are lively and intelligent. Filled with fun and fond of amusement. They have to open a ticket simply to get the number?They should go through the process to ensure that they have attempted all selfhelp options and have used any webforms and reviewed any articles regarding issue before calling. This can help ensure that callers are calling for issues that can only be resolved via speaking with a representative..<ul>
 
 
Every vector space ''V'' with seminorm ''p''('''v''') induces a normed space ''V/W'', called the [[quotient space (linear algebra)|quotient space]], where ''W'' is the subspace of ''V''  consisting of all vectors '''v''' in ''V'' with ''p''('''v''') = 0.   The induced norm on ''V/W'' is clearly well-defined and is given by:
  <li>[http://yaobuy.com.cn/news/html/?197554.html http://yaobuy.com.cn/news/html/?197554.html]</li>
: ''p''(''W'' + '''v''') = ''p''('''v''').
 
 
  <li>[http://sysang.egloos.com/6357210/ http://sysang.egloos.com/6357210/]</li>
Two norms (or seminorms) ''p'' and ''q'' on a vector space ''V'' are '''equivalent''' if there exist two real constants ''c'' and ''C'', with {{nowrap|''c'' > 0}} such that
 
:for every vector '''v''' in ''V'', one has that: {{nowrap|  ''c'' ''q''('''v''') &le; ''p''('''v''') &le; ''C'' ''q''('''v''')}}.
  <li>[http://www.xiongmaoche.com/bbs/forum.php?mod=viewthread&tid=1860403 http://www.xiongmaoche.com/bbs/forum.php?mod=viewthread&tid=1860403]</li>
 
 
A [[topological vector space]] is called '''normable''' ('''seminormable''') if the [[topology]] of the space can be induced by a norm (seminorm).
  <li>[http://www.wikimusicmemorabilia.com/User:Ajqjcvht#Christian_Louboutin_Sydney_weed_control http://www.wikimusicmemorabilia.com/User:Ajqjcvht#Christian_Louboutin_Sydney_weed_control]</li>
 
 
==Notation==
  <li>[http://www.mxw1688.com/news/html/?140776.html http://www.mxw1688.com/news/html/?140776.html]</li>
 
 
If a norm {{nowrap|''p'': ''V'' → '''R'''}} is given on a vector space ''V'' then the norm of a vector '''v''' ∈ ''V'' is usually denoted by enclosing it within double vertical lines: &#x2016;'''v'''&#x2016; := ''p''('''v'''). Such notation is also sometimes used if ''p'' is only a seminorm.
</ul>
 
For the length of a vector in Euclidean space (which is an example of a norm, as [[#Euclidean norm|explained below]]), the notation |'''v'''| with single vertical lines is also widespread.
 
In [[Unicode]], the codepoint of the "double vertical line"<!-- Unicode name --> character &#x2016; is U+2016. The double vertical line should not be confused with the "parallel to" symbol, Unicode U+2225 (&nbsp;&#x2225;&nbsp;). This is usually not a problem because the former is used in parenthesis-like fashion, whereas the latter is used as an [[Infix notation|infix operator]]. The single vertical line | is called "vertical line"<!-- Unicode name --> in Unicode and its codepoint is U+007C.
 
==Examples==
* All norms are seminorms.
* The ''trivial seminorm'' has ''p''('''x''') = 0 for all '''x''' in ''V''.
* The [[absolute value]] is a norm on the [[real numbers]].
* Every [[linear form]] ''f'' on a vector space defines a seminorm by '''x''' → |''f''('''x''')|.
 
===Euclidean norm===<!-- [[L2 norm]] and [[L2 distance]] redirect here -->
{{Main|Euclidean distance}}
On an ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>, the intuitive notion of length of the vector '''''x''''' = (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) is captured by the formula
:<math>\|\boldsymbol{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}.</math>
This gives the ordinary distance from the origin to the point '''''x''''', a consequence of the [[Pythagorean theorem]].
The Euclidean norm is by far the most commonly used norm on '''R'''<sup>''n''</sup>, but there are other norms on this vector space as will be shown below. However all these norms are equivalent in the sense that they all define the same topology.
 
On an ''n''-dimensional [[complex space]] '''C'''<sup>''n''</sup> the most common norm is
:<math>\|\boldsymbol{z}\| := \sqrt{|z_1|^2 + \cdots + |z_n|^2}= \sqrt{z_1 \bar z_1 + \cdots + z_n \bar z_n}.</math>
 
In both cases we can also express the norm as the [[square root]] of the [[inner product]] of the vector and itself:
:<math>\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x}^* \boldsymbol{x}},</math>
where '''''x''''' is represented as a [[column vector]] ([''x''<sub>1</sub>; ''x''<sub>2</sub>; ...; ''x''<sub>''n''</sub>]), and '''''x'''''* denotes its [[conjugate transpose]].
 
This formula is valid for any [[inner product space]], including Euclidean and complex spaces. For Euclidean spaces, the inner product is equivalent to the [[dot product]]. Hence, in this specific case the formula can be also written with the following notation:
:<math>\|\boldsymbol{x}\| := \sqrt{\boldsymbol{x} \cdot \boldsymbol{x}}.</math>
 
The Euclidean norm is also called the '''Euclidean length''', '''''L''<sup>2</sup> distance''', '''ℓ<sup>2</sup> distance''', '''''L''<sup>2</sup> norm''', or  '''ℓ<sup>2</sup> norm'''; see [[Lp space|''L''<sup>''p''</sup> space]].
 
The set of vectors in '''R'''<sup>''n+1''</sup> whose Euclidean norm is a given positive constant forms an [[n-sphere|''n''-sphere]].
 
====Euclidean norm of a complex number====
The Euclidean norm of a [[complex number]] is the [[absolute value#Complex numbers|absolute value]] (also called the '''modulus''') of it, if the [[complex plane]] is identified with the [[Euclidean plane]] '''R'''<sup>2</sup>. This identification of the complex number '''x + ''i''y''' as a vector in the Euclidean plane, makes the quantity <math>\sqrt{x^2 +y^2}</math> (as first suggested by Euler) the Euclidean norm associated with the complex number.
 
===Taxicab norm or Manhattan norm ===
{{Main|Taxicab geometry}}
 
:<math>\|\boldsymbol{x}\|_1 := \sum_{i=1}^{n} |x_i|.</math>
The name relates to the distance a taxi has to drive in a rectangular [[street grid]] to get from the origin to the point ''x''.
 
The set of vectors whose 1-norm is a given constant forms the surface of a [[cross polytope]] of dimension equivalent to that of the norm minus 1.  The Taxicab norm is also called the '''''L''<sub>1</sub> norm'''. The distance derived from this norm is called the [[Manhattan distance]] or '''''L''<sub>1</sub> distance'''.
 
The 1-norm is simply the sum of the absolute values of the columns.
 
In contrast,
 
:<math>\sum_{i=1}^{n} x_i</math>
 
is not a norm because it may yield negative results.
 
===''p''-norm===
{{Main|Lp space|l1=L<sup>p</sup> space}}
 
Let ''p'' ≥ 1 be a real number.
:<math>\|\mathbf{x}\|_p := \bigg( \sum_{i=1}^n |x_i|^p \bigg)^{1/p}.</math>
Note that for ''p'' = 1 we get the [[#Taxicab norm or Manhattan norm|taxicab norm]], for ''p'' = 2 we get the [[#Euclidean norm|Euclidean norm]], and as ''p'' approaches <math>\infty</math> the ''p''-norm approaches the [[Uniform norm|infinity norm]] or [[#Maximum_norm_.28special_case_of:_infinity_norm.2C_uniform_norm.2C_or_supremum_norm.29|maximum norm]].
 
This definition is still of some interest for 0 < ''p'' < 1, but the resulting function does not define a norm,<ref>Except in '''R'''<sup>1</sup>, where it coincides with the Euclidean norm, and '''R'''<sup>0</sup>, where it is trivial.</ref> because it violates the [[triangle inequality]]. What is true for this case of 0 < ''p'' < 1, even in the measurable analog, is that the corresponding ''L<sup>p</sup>'' class is a vector space, and it is also true that the function
:<math>\int_X|f(x)-g(x)|^p d\mu\,\!</math><!-- The \,\! is to keep the formula rendered as PNG instead of HTML. Please don't remove it. -http://meta.wikimedia.org/wiki/Help:Formula -->
(without ''p''th root)  defines a distance that makes ''L''<sup>''p''</sup>(''X'') into a complete metric [[topological vector space]]. These spaces are of great interest in [[functional analysis]], [[probability theory]], and [[harmonic analysis]].
However, outside trivial cases, this topological vector space is not locally convex and has no continuous nonzero linear forms. Thus the topological dual space contains only the zero functional.
 
The derivative of the ''p''-norm is given by
:<math>\frac{\partial}{\partial x_k} \|\mathbf{x}\|_p = \frac{x_k |x_k|^{p-2}}{\|\mathbf{x}\|_{p}^{p-1}}.</math>
 
For the special case of ''p'' = 2, this becomes
:<math>\frac{\partial}{\partial x_k} \|\mathbf{x}\|_2 = \frac{x_k}{\|\mathbf{x}\|_2}</math>,
or
:<math>\frac{\partial}{\partial \mathbf{x}} \|\mathbf{x}\|_2 = \frac{\mathbf{x}}{\|\mathbf{x}\|_2}.</math>
 
===Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)===
[[Image:Vector norm sup.svg|frame|right|<math>\|x\|_\infty = 1</math>]]
{{Main|Maximum norm}}
 
:<math>\|\mathbf{x}\|_\infty := \max \left(|x_1|, \ldots ,|x_n| \right).</math>
 
The set of vectors whose infinity norm is a given constant, ''c'', forms the surface of a [[hypercube]] with edge length 2''c''.
 
=== Zero norm ===
 
In probability and functional analysis, the zero norm induces a complete metric topology for the space of measureable functions and for the [[F-space]] of sequences with [[F-space|F–norm]] <math>(x_n) \mapsto \sum_n{2^{-n} x_n/(1+x_n)}</math>, which is discussed by Stefan Rolewicz in ''Metric Linear Spaces''.<ref name="RolewiczControl">{{Citation | title=Functional analysis and control theory: Linear systems|last=Rolewicz |first=Stefan|year=1987| isbn=90-277-2186-6| publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers|oclc=13064804|edition=Translated from the Polish by Ewa Bednarczuk|series=Mathematics and its Applications (East European Series)|location=Dordrecht; Warsaw|volume=29|pages=xvi+524|mr=920371}}</ref>
 
====Hamming distance of a vector from zero====
 
{{See also|Hamming distance|discrete metric}}
In [[metric geometry]], the [[discrete metric]] takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the ''[[Hamming distance]]'', which is important in [[coding theory|coding]] and [[information theory]].  In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero. However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness. When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.
 
In [[signal processing]] and [[statistics]], [[David Donoho]] referred to the ''zero'' '''"'''''norm'''''"''' with quotation marks. Following Donoho's notation, the zero "norm" of '''x''' is simply the number of non-zero coordinates of '''x''', or the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of ''p''-norms as ''p'' approaches 0.  Of course, the zero "norm" is '''not''' a B-norm, because it is not [[Homogeneous_function#Positive_homogeneity|positive homogeneous]]. It is not even an F-norm, because it is discontinuous, jointly and severally, with respect to the scalar argument in scalar-vector multiplication and with respect to its vector argument. [[Abuse of terminology|Abusing terminology]], some engineers omit Donoho's quotation marks and inappropriately call the number-of-nonzeros function the L0 norm (sic.), also misusing the notation for the [[Lp space|Lebesgue space]] of [[measurable function]]s.
 
===Other norms===
Other norms on '''R'''<sup>''n''</sup> can be constructed by combining the above; for example
:<math>\|x\| := 2|x_1| + \sqrt{3|x_2|^2 + \max(|x_3|,2|x_4|)^2}</math>
is a norm on '''R'''<sup>4</sup>.
 
For any norm and any injective [[linear transformation]] ''A'' we can define a new norm of ''x'', equal to
:<math>\|Ax\|.</math>
In 2D, with ''A'' a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. In 2D, each ''A'' applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a [[parallelogram]] of a particular shape, size and orientation. In 3D this is similar but different for the 1-norm ([[octahedron]]s) and the maximum norm ([[Prism (geometry)|prism]]s with parallelogram base).
 
All the above formulas also yield norms on '''C'''<sup>''n''</sup> without modification.
 
=== Infinite-dimensional case ===
The generalization of the above norms to an infinite number of components leads to the [[Lp space|''L''<sup>''p''</sup> space]]s, with norms
 
:<math> \|x\|_p = \bigg(\sum_{i\in\mathbb N}|x_i|^p\bigg)^{1/p} \text{ resp. } \|f\|_{p,X} = \bigg(\int_X|f(x)|^p\,\mathrm dx\bigg)^{1/p} </math>
 
(for complex-valued sequences ''x'' resp. functions ''f'' defined on <math>X\subset\mathbb R</math>), which can be further generalized (see [[Haar measure]]).
 
Any [[inner product]] induces in a natural way the norm <math>\|x\| := \sqrt{\langle x,x\rangle}.</math>
 
Other examples of infinite dimensional normed vector spaces can be found in the [[Banach space]] article.
 
== Properties ==
<!--[[Equivalent norms]] redirects here-->
[[Image:Vector norms.svg|frame|right|Illustrations of [[unit circle]]s in different norms.]]
 
The concept of [[unit circle]] (the set of all vectors of norm 1) is different in different norms: for the 1-norm the unit circle in '''R'''<sup>2</sup> is a [[square (geometry)|square]], for the 2-norm (Euclidean norm) it is the well-known unit [[circle]], while for the infinity norm it is a different square.  For any ''p''-norm it is a [[superellipse]] (with congruent axes).  See the accompanying illustration. Note that due to the definition of the norm, the unit circle is always [[convex set|convex]] and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle).
 
In terms of the vector space, the seminorm defines a [[topology]] on the space, and this is a [[Hausdorff space|Hausdorff]] topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A [[sequence]] of vectors <math>\{v_n\}</math> is said to [[modes of convergence|converge]] in norm to <math>v</math> if <math>\|v_n - v\| \rightarrow 0</math> as <math>n \to \infty</math>. Equivalently, the topology consists of all sets that can be represented as a union of open [[ball (mathematics)|balls]].
 
Two norms ||•||<sub>α</sub> and ||•||<sub>β</sub> on a vector space ''V'' are called ''equivalent'' if there exist positive real numbers ''C'' and ''D'' such that
:<math>C\|x\|_\alpha\leq\|x\|_\beta\leq D\|x\|_\alpha</math>
for all ''x'' in ''V''. For instance, on <math>\mathbf{C}^n</math>, if ''p'' > ''r'' > 0, then
:<math>\|x\|_p \leq \|x\|_r \leq n^{(1/r-1/p)}\|x\|_p.</math>
 
In particular,
:<math>\|x\|_2\le\|x\|_1\le\sqrt{n}\|x\|_2</math>
:<math>\|x\|_\infty\le\|x\|_2\le\sqrt{n}\|x\|_\infty</math>
:<math>\|x\|_\infty\le\|x\|_1\le n\|x\|_\infty.</math>
 
If the vector space is a finite-dimensional real/complex one, all norms are equivalent.
On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.
 
Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space is [[uniformly isomorphic]].
 
Every (semi)-norm is a [[sublinear function]], which implies that every norm is a [[convex function]]. As a result, finding a global optimum of a norm-based [[objective function]] is often tractable.
 
Given a finite family of seminorms ''p''<sub>''i''</sub> on a vector space the sum
:<math>p(x):=\sum_{i=0}^n p_i(x)</math>
is again a seminorm.
 
For any norm ''p'' on a vector space ''V'', we have that for all '''u''' and '''v''' ∈ ''V'':
:p('''u''' ± '''v''') ≥ | p('''u''') − p('''v''') |
 
If <math>X</math> and <math>Y</math> are normed spaces and <math>u : X \to Y</math> is a continuous linear map, then the norm of <math>u</math> and the norm of the [[transpose]] of <math>u</math> are equal.<ref>Treves pp. 242 - 243</ref>
 
For the [[Lp space|l<sup>p</sup>]] norms, we have [[Hölder's inequality]]<ref name="GOLUB">{{cite book|last=Golub|first=Gene|authorlink=Gene_H._Golub|coauthors=Charles F. Van Loan|title=Matrix Computations - Third Edition|publisher=The Johns Hopkins University Press|year=1996|location=Baltimore|isbn=0-8018-5413-X|page=53}}</ref>
:<math>|x^\mathsf{T} y|\le\| x\|_p\|y\|_q\qquad \frac{1}{p}+\frac{1}{q}=1.</math>
A special case of this is the [[Cauchy–Schwarz inequality]]:<ref name="GOLUB"/>
:<math>|x^\mathsf{T} y|\le\|x\|_2\|y\|_2.</math>
 
== Classification of seminorms: Absolutely convex absorbing sets ==
All seminorms on a vector space ''V'' can be classified in terms of [[absolutely convex]] [[absorbing set]]s in ''V''.  To each such set, ''A'', corresponds a seminorm ''p<sub>A</sub>'' called the '''[[Minkowski functional|gauge]]''' of ''A'', defined as
:''p<sub>A</sub>''(''x'') := [[infimum|inf]]{α : α > 0, ''x'' ∈ α ''A''}
with the property that
:{''x'' : ''p<sub>A</sub>''(''x'') < 1} ⊆ ''A'' ⊆ {''x'' : ''p<sub>A</sub>''(''x'') ≤ 1}.
Conversely:
 
Any [[locally convex topological vector space]] has a [[local basis]] consisting of absolutely convex sets. A common method to construct such a basis is to use a family (p) of seminorms p that [[separating|separates points]]: the collection of all finite intersections of sets {p<1/n} turns the space into a [[locally convex topological vector space]] so that every p is [[Continuous function|continuous]].
 
Such a method is used to design [[Weak topology|weak and weak* topologies]].
 
norm case:
:Suppose now that (p) contains a single p: since (p) is [[separating]], p is a norm, and ''A''={p<1} is its open [[unit ball]]. Then ''A'' is an absolutely convex [[bounded set|bounded]] neighbourhood of 0, and {{nowrap|1=''p'' = ''p<sub>A</sub>''}} is continuous.
 
:The converse is due to [[Kolmogorov]]: any locally convex and locally bounded topological vector space is [[normable]]. Precisely:
:If V is an absolutely convex bounded neighbourhood of 0, the gauge g<sub>V</sub> (so that V={g<sub>V</sub> <1}) is a norm.
 
==Generalizations==
 
There are several generalizations of norms and semi-norms. If ''p'' is absolute homogeneity but in place of subadditivity we require that
{|
|-
| align="right" |  2'. || there is a <math>b \geq 1</math> such that <math>p(u + v) \leq b(p(u) + p(v))</math> for all <math>u, v \in V</math>
|}
then ''p'' satisfies the triangle inequality butis called a [[quasinorm|'''quasi-seminorm''']] and the smallest value of ''b'' for which this holds is called the '''multiplier of ''p'''''; if in addition ''p'' separates points then it is called a '''quasi-norm'''.
 
On the other hand, if ''p'' satisfies the triangle inequality but but in place of absolute homogeneity we require that
{|
|-
| align="right" |  1'. || there exists a ''k'' such that <math>0 < k \leq 1</math> and for all <math>v \in V</math> and scalars <math>\lambda</math> <math>p(\lambda v) = | \lambda |^k p(v)</math>
|}
then ''p'' is called a '''''k''-seminorm'''.
 
We have the following relationship between quasi-seminorms and ''k''-seminorms:
: Suppose that ''q'' is a quasi-seminorm on a vector space ''X'' with multiplier ''b''. If <math>0 < k < log^2_2 b</math> then there exists ''k''-seminorm ''p'' on ''X'' equivalent to ''q''.
 
== See also ==
*[[Asymmetric norm]], a generalization of a norm for which ||''x''|| and ||&minus;''x''|| are not necessarily equal.
*[[Berkovich space]], a topological space whose points are the multiplicative seminorms on a ring
*[[Gowers norm]] in [[additive combinatorics]]
*[[Mahalanobis distance]]
*[[Manhattan distance]]
*[[Matrix norm]]
*[[Normed vector space]]
*[[One-form]]
*[[Relation of norms and metrics]], a translation invariant and homogeneous metric can be used to define a norm.
 
== Notes ==
<!--<nowiki>
See http://en.wikipedia.org/wiki/Wikipedia:Footnotes for an explanation of how to generate footnotes using the <ref> and </ref> tags, and the template below.
</nowiki>-->
<references/>
 
== References ==
{{refbegin}}
* {{cite book
  | last = Bourbaki | first = Nicolas
  | title = Topological vector spaces
  | year = 1987
  | chapter = Chapters 1–5
  | publisher = [[Springer Science+Business Media|Springer]]
  | isbn = 3-540-13627-4
  | ref = harv
  }}
* {{cite book
  | last = Prugovečki | first = Eduard
  | title = Quantum mechanics in Hilbert space
  | year = 1981 | edition = 2nd
  | publisher = Academic Press
  | url =  |page=20
  | isbn = 0-12-566060-X
  | ref = harv
  }}
* {{cite book | last = Tr\`{e}ves | first = Fran\c{c}ois | title=Topological Vector Spaces, Distributions and Kernels | publisher=[[Academic Press, Inc.]] | year=1995 | isbn=0-486-45352-9 | pages=136-149, 195-201, 240-252, 335-390, 420-433 }}
{{refend}}
* {{cite book | author=S.M. Khaleelulla | title=Counterexamples in Topological Vector Spaces | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=936 | date=1982 | isbn=978-3-540-11565-6 | pages=3-5}}
 
{{Functional Analysis}}
 
{{DEFAULTSORT:Norm (Mathematics)}}
<!--Categories-->
[[Category:Norms (mathematics)| ]]
[[Category:Linear algebra]]

Latest revision as of 07:15, 27 October 2014

Nike Free Run for that religious bunch

Besides that, it's about 15 megabytes. I'm the only other person that has it, besides Carlos. I recieve it. Nonetheless the conflict and dysfunction among the crew of SGU was still true to life and closer to adulthood than te Stargate franchise will ever get.

Making a new record each time a coverage change occurs recording the start and ending dates and times of each plan storing the historical records in a history table unioning the historical records and also the current records together to obtain a full report Figure 2 shows a business relationship diagram (ERD) that illustrates my suggestions. The diagram is really a basic model of the solution; to prevent confusion, I've minimized the attribute set.

I always look forward to reading your newsletters and e-mail to see what new project your chapter volunteers are planning. Although I live 2 hours away, I have made several trips for your meetings for your excellent presentations. You can also indicate the person you are looking for here if you want to find their postcode. Just type the name of the person.

Seven things I learned at IITP 2013. The summary of this year's Institute of IT Nike Free Run Professionals 2013 Conference was first published last weekend and was popular enough to entitled to the previous week's top stories as well as heading this week's list.

Lets get it done this way: If all the species of the planet relied solely on homosexual acts the world would have no life, who have been better for the planet. Or, for the religious bunch: If God (when there Abercrombie Australia is a God) created Adam and Eve how come they are of different genders? Answer: God created Adam and Eve for reproduction to be able to procreate the planet with unhappy people.

Planning Status R Garage courts are in good condition. Retain parking use Garage courts are in good condition. Next Mass Effect Serves as Sequel to Main Trilogy, New Races Get Details Report Download UC Browser 9.3 for Java Download BBM for Android 1.0.2.83 Who Needs Winamp When you are able Play Music from the Terminal Download UC Browser 9.2 for Java 14 Alleged Members of Anonymous and RedHack Referred to Ankara Court Top Five Microsoft Funny Quotes About Oakley Holbrook Linux Samsung Galaxy Grand 2 Goes Official with DualSIM Support and 5.25Inch Display Windows 9 Concepts: Designers Think of Nike Air Max 2013 the Next Microsoft Operating System Microsoft Confirms Windows 9 Could Merge Modern OS VersionsScribd, the popular document sharing site has announced a new round of funding, raising $13 million from a number of institutional investors. This is the company third funding round and it plans to use the money to fund growth, by hiring several new people, mainly in the engineering department.Scribd enables users to upload and share documents with other users or make them completely public, the same as Flickr or YouTube for documents.MLC Investments and SVB Capital led the round with existing investors Redpoint Ventures, Charles River Ventures and Kinsey Hills Group also participating.It last funding round closed a couple of years ago, it raised $9 million in December 2008.

Oakley Sunglasses Cheap

Products for Geeks, Gamers, and General Users The brand new site is segmented into eight distinct product categories for all sorts of PC and Mac users: Memory: the easiest to install and most costeffective performance upgrade, adding more memory can dramatically Oakley Sunglasses Cheap improve app and system responsiveness, gaming, and specific memory intensive usages for example video streaming over wireless networks. Graphics Cards: gamers looking for the ultimate experience can choose from a variety Nike Air Max 90 of options including the R9 290X, the world's fastest graphics card.

Since his immunity system starts to dysfunction he would need extra. If you want to know about the nutrition factor of the AIDS patient, visit the site. A little vending machine business could be started on quite a low budget therefore it is better to start out slowly and face a smaller amount of risk. After that you can choose to invest further to some great extent as soon as the business starts to earn a certain amount of income, and after you determine that vending is really the business that you desire to do..

Deuces Cracked have place a lot of effort into developing a well stocked catalog of full ring quite happy with over 120 videos already available. Furthermore, new content is produced on a weekly basis by full ring specialists such as newmanmi, bottomset, BalugaWhale, goldseraph, RapidEvolution and DjSensei.

Mucha has been hired as Managing Editor in charge of correspondents. Mucha will oversee GlobalPost' corps of correspondents, multimedia, and can contribute to planning and executing special features and reporting projects. "I stopped taking them. A part of that created in me this nervous anxiety, and that is when I was really screwed.

Not to mention that its not that many guys. Have a small break to cool down after which check out the steam room (parilka), where one can be beaten about the body with dried branches (vennik), usually birch, oak or juniper. Thirteen years later the S 500 has still not exceeded that 2000 high by a decisive margin. For this type of breakout to occur, it would need to break not only the 2000 high but also the marginally higher record of 2007, by a number of percentage points.

I know that Air Max Thea Australia in primary he didn't like to say anything or ask anything just in case he was wrong or got laughed at (bullying issue) but i'd really love it if he felt Nike Air Max Australia able to be himself more. He is clever, funny and generally a nice lad.. Some minisites converts constantly while others get only occasional sales. What makes such a huge difference? It is something related to design layout.

Beats Headphones Australia huts

Please configure your browser or system settings to permit popups on CSUMentor. Federal Student Aid Beats Headphones Australia provides publications, online tools, Ray Ban Sunglasses Australia videos, and other resources to help you prepare and purchase college or career school. Forces are getting stronger. And they are getting ready for more, he said.

For one thing, he wading into volatile territory having a story set in the Middle East. You on really tricky political ground, and also you know no matter what you do, people have strongly held beliefs which will lead them to see the material through one lens or another, you just have to try to adhere really rigidly to the facts, he says..

""Ye are the light around the globe. A city that is set on an hill cannot be hid. Topic galleries provide easy access to stories and photos about people, places, organizations, events and subjects of great interest to you. They bring together rich.. No training required. Lots of opportunities.

I don think that they have much to gain beyond having a deterrent [against any attack]. But the fear is that they will export nuclearweapons materials or technology, therefore we can ignore them either. If this sounds like too much there's an online design service (pounds 40) in which you send in detailed plans of the garden, what you want, and how much you really can afford. Expert Nicholas Roeber will tell you what to do.

Trustcash's President and Chief Executive Officer, Kent Carasquero, stated, "Using the best technology to support our new website must go hand in hand with bringing the Trustcash brand alive by infusing it with a provocative voice that is both distinct and compelling. Our goal would be to create a website that hits the mark for both the online and debit card clients.

At first Violet whistled him, correctly, for offensive interference and negated the basket. Then she changed her mind when the Knicks argued that Nate did not touch the ball (so it appeared he did not).. You mustn't misrepresent your relationship with St. Jude or present false or misleading impressions about St.

By the Air Max Australia way, Im all for biting the hand Nike Air Max 2014 that fed me. Angelfire blows. Roll it over the area you wish to cover and stake the edges using landscape stakes, that also is found at garden centers. Heavy landscape fabric also is effective when thrown over a compost heap.

Your main income and resources must come from what the game provides (trade specials, good starting locations, huts, etc.) and from interaction with the AI, by trading techs, and becoming tribute and/or gifts. You can't possibly do it all on your own..

Ralph Lauren Australia who's in police custody

Although the serial Ralph Lauren Australia number isn't externally listed, but it is available when plugged into a computer on the "Summary" tab in iTunes. If the display works enough to use the screen, the serial number also can be viewed by browsing Settings > General > About from inside the iOS.

Once you succeed locally, you succeed nationally,' says Gupta. Remember Aaj Tak? It began as a beacon of hope for small , local advertisers. In spite of its more undesired flavor, robusta beans are confirmed to be an organic pesticide and stimulant, also growing in places that the Arabica beans can not grow. Therefore, it's used as an inexpensive substitute for Arabica in commercial coffee blends and in nearly every instant coffee brand..

U can also, with practice. Help p lvl someone. Having the bible text and notes in separate frames so the bible text can be read without distraction.6. The alignment from the bible text and the notes. Oakley Holbrook Without these any machine is just a piece of metal or plastic. There are different types of software available for different uses.

This press release may be deemed to contain forwardlooking information. Any forwardlooking statements are made pursuant to the safe harbor provisions from the Private Securities Litigation Reform Act The non-public Securities Litigation Reform Act of 1995 (PSLRA) implemented several significant Mont Blanc Australia substantive changes affecting some instances brought under the federal securities laws, including changes associated with pleading, discovery, liability, class representation and awards fees as well as 1995.

It is absorbed through the skin and into the bloodstream. It can benefit your body keep estrogen and progesterone levels in balance, leading to maintained sense of equilibrium.. Her neck had bruise marks suggesting that attempts were made to strangle her."The rape accused Manoj, who is in police custody, had also raped his wife before marriage. The village panchayat had then ordered Manoj to marry the lady.

In my own experience, people can read as much Beats Headphones Australia as they want about something in advance, but it'll all venture out the window as soon as they do it for real. They'll still make a load of newbie mistakes once they try applying the skills they read about.

This is the basis of noise canceling, enabling you to escape your surroundings and listen to what you want. Sometimes however, this comes confined sticker price. Generally people are lively and intelligent. Filled with fun and fond of amusement. They have to open a ticket simply to get the number?They should go through the process to ensure that they have attempted all selfhelp options and have used any webforms and reviewed any articles regarding issue before calling. This can help ensure that callers are calling for issues that can only be resolved via speaking with a representative..

Retrieved from "https://en.formulasearchengine.com/index.php?title=Gramian_matrix&oldid=235580"