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{{About|linear algebra and analysis|field theory|Field norm|ideals|Ideal norm|group theory|Norm (group)|norms in descriptive set theory|prewellordering}}
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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).
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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]].
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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.
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==Definition==
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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'',
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# ''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]]'').
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# If ''p''('''v''') = 0 then '''v''' is the [[zero vector]] (''separates points'').
 
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By the first axiom, absolute homogeneity, we have ''p''('''0''') = 0 and ''p''('''-v''') = ''p''('''v'''), so that by the triangle inequality
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: ''p''('''v''') ≥ 0 (''positivity'').


A '''seminorm''' is a '''norm''' with the 3rd property (separating points) removed.
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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:
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: ''p''(''W'' + '''v''') = ''p''('''v''').
 
 
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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''')}}.
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A [[topological vector space]] is called '''normable''' ('''seminormable''') if the [[topology]] of the space can be induced by a norm (seminorm).
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==Notation==
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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 06:15, 27 October 2014

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