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Yes, I’ll donate $2 No, but maybe later when I have more time Norm (mathematics) From Wikipedia, the free encyclopedia Jump to navigationJump to search This article is about norms of normed vector spaces. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.[1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.
The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".[1] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "{\displaystyle ,\leq ,}{\displaystyle ,\leq ,}" in the homogeneity axiom.[2] It can also refer to a norm that can take infinite values,[3] or to certain functions parametrised by a directed set.[4]
Contents 1 Definition 1.1 Equivalent norms 1.2 Notation 2 Examples 2.1 Absolute-value norm 2.2 Euclidean norm 2.3 Quaternions and octonions 2.4 Finite-dimensional complex normed spaces 2.5 Taxicab norm or Manhattan norm 2.6 p-norm 2.7 Maximum norm (special case of: infinity norm, uniform norm, or supremum norm) 2.8 Zero norm 2.9 Infinite dimensions 2.10 Composite norms 2.11 In abstract algebra 3 Properties 3.1 Equivalence 4 Classification of seminorms: absolutely convex absorbing sets 5 See also 6 References 7 Bibliography Definition Given a vector space {\displaystyle X}X over a subfield {\displaystyle F}F of the complex numbers {\displaystyle \mathbb {C} ,}{\displaystyle \mathbb {C} ,} a norm on {\displaystyle X}X is a real-valued function {\displaystyle p:X\to \mathbb {R} }{\displaystyle p:X\to \mathbb {R} } with the following properties, where {\displaystyle |s|}{\displaystyle |s|} denotes the usual absolute value of a scalar {\displaystyle s}s:[5]
Subadditivity/Triangle inequality: {\displaystyle p(x+y)\leq p(x)+p(y)}{\displaystyle p(x+y)\leq p(x)+p(y)} for all {\displaystyle x,y\in X.}{\displaystyle x,y\in X.} Absolute homogeneity: {\displaystyle p(sx)=\left|s\right|p(x)}{\displaystyle p(sx)=\left|s\right|p(x)} for all {\displaystyle x\in X}x\in X and all scalars {\displaystyle s.}s. Positive definiteness/Point-separating: for all {\displaystyle x\in X,}{\displaystyle x\in X,} if {\displaystyle p(x)=0}p(x) = 0 then {\displaystyle x=0.}x=0. Because property (2.) implies {\displaystyle p(0)=0,}{\displaystyle p(0)=0,} some authors replace property (3.) with the equivalent condition: for every {\displaystyle x\in X,}{\displaystyle x\in X,} {\displaystyle p(x)=0}p(x) = 0 if and only if {\displaystyle x=0.}x=0. A seminorm on {\displaystyle X}X is a function {\displaystyle p:X\to \mathbb {R} }{\displaystyle p:X\to \mathbb {R} } that has properties (1.) and (2.)[6] so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if {\displaystyle p}p is a norm (or more generally, a seminorm) then {\displaystyle p(0)=0}{\displaystyle p(0)=0} and that {\displaystyle p}p also has the following property:
Non-negativity: {\displaystyle p(x)\geq 0}{\displaystyle p(x)\geq 0} for all {\displaystyle x\in X.}{\displaystyle x\in X.} Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
Equivalent norms Suppose that {\displaystyle p}p and {\displaystyle q}q are two norms (or seminorms) on a vector space {\displaystyle X.}X. Then {\displaystyle p}p and {\displaystyle q}q are called equivalent, if there exist two positive real constants {\displaystyle c}c and {\displaystyle C}C with {\displaystyle c>0}c>0 such that for every vector {\displaystyle x\in X,}{\displaystyle x\in X,}
{\displaystyle cq(x)\leq p(x)\leq Cq(x).}{\displaystyle cq(x)\leq p(x)\leq Cq(x).} The relation "{\displaystyle p}p is equivalent to {\displaystyle q}q" is reflexive, symmetric ({\displaystyle cq\leq p\leq Cq}{\displaystyle cq\leq p\leq Cq} implies {\displaystyle {\tfrac {1}{Cp\leq q\leq {\tfrac {1}{cp}{\displaystyle {\tfrac {1}{Cp\leq q\leq {\tfrac {1}{cp}), and transitive and thus defines an equivalence relation on the set of all norms on {\displaystyle X.}X. The norms {\displaystyle p}p and {\displaystyle q}q are equivalent if and only if they induce the same topology on {\displaystyle X.}X.[7] Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.[7] Notation If a norm {\displaystyle p:X\to \mathbb {R} }{\displaystyle p:X\to \mathbb {R} } is given on a vector space {\displaystyle X,}X, then the norm of a vector {\displaystyle z\in X}z\in X is usually denoted by enclosing it within double vertical lines: {\displaystyle |z|=p(z).}{\displaystyle |z|=p(z).} Such notation is also sometimes used if {\displaystyle p}p is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation {\displaystyle |x|}|x| with single vertical lines is also widespread.
Examples Every (real or complex) vector space admits a norm: If {\displaystyle x_{\bullet }=\left(x_{i}\right){i\in I}}{\displaystyle x{\bullet }=\left(x_{i}\right){i\in I}} is a Hamel basis for a vector space {\displaystyle X}X then the real-valued map that sends {\displaystyle x=\sum {i\in I}s{i}x{i}\in X}{\displaystyle x=\sum {i\in I}s{i}x_{i}\in X} (where all but finitely many of the scalars {\displaystyle s_{i}}s_{i} are {\displaystyle 0}{\displaystyle 0}) to {\displaystyle \sum {i\in I}\left|s{i}\right|}{\displaystyle \sum {i\in I}\left|s{i}\right|} is a norm on {\displaystyle X.}X.[8] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
Absolute-value norm The absolute value
{\displaystyle |x|=|x|}{\displaystyle |x|=|x|} is a norm on the one-dimensional vector spaces formed by the real or complex numbers. Any norm {\displaystyle p}p on a one-dimensional vector space {\displaystyle X}X is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces {\displaystyle f:\mathbb {F} \to X,}{\displaystyle f:\mathbb {F} \to X,} where {\displaystyle \mathbb {F} }\mathbb {F} is either {\displaystyle \mathbb {R} }\mathbb {R} or {\displaystyle \mathbb {C} ,}{\displaystyle \mathbb {C} ,} and norm-preserving means that {\displaystyle |x|=p(f(x)).}{\displaystyle |x|=p(f(x)).} This isomorphism is given by sending {\displaystyle 1\in \mathbb {F} }{\displaystyle 1\in \mathbb {F} } to a vector of norm {\displaystyle 1,}1, which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.
Euclidean norm Further information: Euclidean norm and Euclidean distance On the {\displaystyle n}n-dimensional Euclidean space {\displaystyle \mathbb {R} ^{n},}{\displaystyle \mathbb {R} ^{n},} the intuitive notion of length of the vector {\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)}{\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)} is captured by the formula[9]
{\displaystyle |{\boldsymbol {x}}|{2}:={\sqrt {x{1}^{2}+\cdots +x_{n}^{2}}}.}{\displaystyle |{\boldsymbol {x}}|{2}:={\sqrt {x{1}^{2}+\cdots +x_{n}^{2}}}.} This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem. This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares.[10]
The Euclidean norm is by far the most commonly used norm on {\displaystyle \mathbb {R} ^{n},}{\displaystyle \mathbb {R} ^{n},}[9] 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.
The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis. Hence, the Euclidean norm can be written in a coordinate-free way as
{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.} The Euclidean norm is also called the {\displaystyle L^{2L^{2} norm,[11] {\displaystyle \ell ^{2\ell ^{2} norm, 2-norm, or square norm; see {\displaystyle L^{pL^{p} space. It defines a distance function called the Euclidean length, {\displaystyle L^{2L^{2} distance, or {\displaystyle \ell ^{2}}\ell ^{2} distance.
The set of vectors in {\displaystyle \mathbb {R} ^{n+1}}{\displaystyle \mathbb {R} ^{n+1}} whose Euclidean norm is a given positive constant forms an {\displaystyle n}n-sphere.
Euclidean norm of complex numbers See also: Dot product § Complex vectors The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane {\displaystyle \mathbb {R} ^{2}.}{\displaystyle \mathbb {R} ^{2}.} This identification of the complex number {\displaystyle x+iy}{\displaystyle x+iy} as a vector in the Euclidean plane, makes the quantity {\textstyle {\sqrt {x^{2}+y^{2}}}}{\textstyle {\sqrt {x^{2}+y^{2}}}} (as first suggested by Euler) the Euclidean norm associated with the complex number.
Quaternions and octonions See also: Quaternion and Octonion There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers {\displaystyle \mathbb {R} ,}{\displaystyle \mathbb {R} ,} the complex numbers {\displaystyle \mathbb {C} ,}{\displaystyle \mathbb {C} ,} the quaternions {\displaystyle \mathbb {H} ,}{\displaystyle \mathbb {H} ,} and lastly the octonions {\displaystyle \mathbb {O} ,}{\displaystyle \mathbb {O} ,} where the dimensions of these spaces over the real numbers are {\displaystyle 1,2,4,{\text{ and }}8,}{\displaystyle 1,2,4,{\text{ and }}8,} respectively. The canonical norms on {\displaystyle \mathbb {R} }\mathbb {R} and {\displaystyle \mathbb {C} }\mathbb{C} are their absolute value functions, as discussed previously.
The canonical norm on {\displaystyle \mathbb {H} }\mathbb {H} of quaternions is defined by
{\displaystyle \lVert q\rVert ={\sqrt {,qq^{}~}}={\sqrt {,q^{}q~={\sqrt {,a^{2}+b^{2}+c^{2}+d^{2}~}{\displaystyle \lVert q\rVert ={\sqrt {,qq^{}~}}={\sqrt {,q^{}q~={\sqrt {,a^{2}+b^{2}+c^{2}+d^{2}~} for every quaternion {\displaystyle q=a+b,\mathbf {i} +c,\mathbf {j} +d,\mathbf {k} }{\displaystyle q=a+b,\mathbf {i} +c,\mathbf {j} +d,\mathbf {k} } in {\displaystyle \mathbb {H} .}{\mathbb {H}}. This is the same as the Euclidean norm on {\displaystyle \mathbb {H} }\mathbb {H} considered as the vector space {\displaystyle \mathbb {R} ^{4}.}{\displaystyle \mathbb {R} ^{4}.} Similarly, the canonical norm on the octonions is just the Euclidean norm on {\displaystyle \mathbb {R} ^{8}.}{\displaystyle \mathbb {R} ^{8}.} Finite-dimensional complex normed spaces On an {\displaystyle n}n-dimensional complex space {\displaystyle \mathbb {C} ^{n},}{\displaystyle \mathbb {C} ^{n},} the most common norm is
{\displaystyle |{\boldsymbol {z}}|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}{1}+\cdots +z{n}{\bar {z}}{n}}}.}{\displaystyle |{\boldsymbol {z}}|:={\sqrt {\left|z{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}{1}+\cdots +z{n}{\bar {z}}_{n}}}.} In this case, the norm can be expressed as the square root of the inner product of the vector and itself:
{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},}{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},} where {\displaystyle {\boldsymbol {x}}}{\boldsymbol {x}} is represented as a column vector {\displaystyle {\begin{bmatrix}x_{1};x_{2};\dots ;x_{n}\end{bmatrix}}^{\rm {T}}}{\displaystyle {\begin{bmatrix}x_{1};x_{2};\dots ;x_{n}\end{bmatrix}}^{\rm {T}}} and {\displaystyle {\boldsymbol {x}}^{H}}{\displaystyle {\boldsymbol {x}}^{H}} denotes its conjugate transpose. This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation:
{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.} Taxicab norm or Manhattan norm Main article: Taxicab geometry {\displaystyle |{\boldsymbol {x}}|{1}:=\sum {i=1}^{n}\left|x{i}\right|.}{\displaystyle |{\boldsymbol {x}}|{1}:=\sum {i=1}^{n}\left|x{i}\right|.} The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point {\displaystyle x.}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 {\displaystyle \ell ^{1}}\ell ^{1} norm. The distance derived from this norm is called the Manhattan distance or {\displaystyle \ell _{1}}\ell _{1} distance.
The 1-norm is simply the sum of the absolute values of the columns.
In contrast,
{\displaystyle \sum {i=1}^{n}x{i}}{\displaystyle \sum {i=1}^{n}x{i}} is not a norm because it may yield negative results. p-norm Main article: Lp space Let {\displaystyle p\geq 1}p\geq 1 be a real number. The {\displaystyle p}p-norm (also called {\displaystyle \ell {p}}{\displaystyle \ell {p}}-norm) of vector {\displaystyle \mathbf {x} =(x{1},\ldots ,x{n})}{\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is[9]
{\displaystyle |\mathbf {x} |{p}:=\left(\sum {i=1}^{n}\left|x{i}\right|^{p}\right)^{1/p}.}{\displaystyle |\mathbf {x} |{p}:=\left(\sum {i=1}^{n}\left|x{i}\right|^{p}\right)^{1/p}.} For {\displaystyle p=1,}{\displaystyle p=1,} we get the taxicab norm, for {\displaystyle p=2}p=2 we get the Euclidean norm, and as {\displaystyle p}p approaches {\displaystyle \infty }\infty the {\displaystyle p}p-norm approaches the infinity norm or maximum norm: {\displaystyle |\mathbf {x} |{\infty }:=\max {i}\left|x{i}\right|.}{\displaystyle |\mathbf {x} |{\infty }:=\max {i}\left|x{i}\right|.} The {\displaystyle p}p-norm is related to the generalized mean or power mean. For {\displaystyle p=2,}{\displaystyle p=2,} the {\displaystyle |,\cdot ,|{2}}{\displaystyle |,\cdot ,|{2}}-norm is even induced by a canonical inner product {\displaystyle \langle ,\cdot ,,\cdot \rangle ,}{\displaystyle \langle ,\cdot ,,\cdot \rangle ,} meaning that {\displaystyle |\mathbf {x} |{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}{\displaystyle |\mathbf {x} |{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} for all vectors {\displaystyle \mathbf {x} .}{\displaystyle \mathbf {x} .} This inner product can expressed in terms of the norm by using the polarization identity. On {\displaystyle \ell ^{2},}{\displaystyle \ell ^{2},} this inner product is the Euclidean inner product defined by
{\displaystyle \langle \left(x_{n}\right){n},\left(y{n}\right){n}\rangle {\ell ^{2}}~=~\sum {n}{\overline {x{n}y{n{\displaystyle \langle \left(x{n}\right){n},\left(y{n}\right)_{n}\rangle {\ell ^{2}}~=~\sum {n}{\overline {x{n}y{n while for the space {\displaystyle L^{2}(X,\mu )}L^{2}(X,\mu ) associated with a measure space {\displaystyle (X,\Sigma ,\mu ),}{\displaystyle (X,\Sigma ,\mu ),} which consists of all square-integrable functions, this inner product is {\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)g(x),\mathrm {d} x.}{\displaystyle \langle f,g\rangle _{L^{2=\int _{X}{\overline {f(x)g(x),\mathrm {d} x.}This definition is still of some interest for {\displaystyle 0<p<1,}{\displaystyle 0<p<1,} but the resulting function does not define a norm,[12] because it violates the triangle inequality. What is true for this case of {\displaystyle 0<p<1,}{\displaystyle 0<p<1,} even in the measurable analog, is that the corresponding {\displaystyle L^{pL^{p} class is a vector space, and it is also true that the function
{\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu }{\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu } (without {\displaystyle p}pth root) defines a distance that makes {\displaystyle L^{p}(X)}{\displaystyle L^{p}(X)} into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic analysis. However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional. The partial derivative of the {\displaystyle p}p-norm is given by
{\displaystyle {\frac {\partial }{\partial x_{k}}}|\mathbf {x} |{p}={\frac {x{k}\left|x_{k}\right|^{p-2}}{|\mathbf {x} |{p}^{p-1}}}.}{\displaystyle {\frac {\partial }{\partial x{k}}}|\mathbf {x} |{p}={\frac {x{k}\left|x_{k}\right|^{p-2}}{|\mathbf {x} |_{p}^{p-1}}}.} The derivative with respect to {\displaystyle x,}x, therefore, is
{\displaystyle {\frac {\partial |\mathbf {x} |{p}}{\partial \mathbf {x} ={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2{|\mathbf {x} |{p}^{p-1}}}.}{\displaystyle {\frac {\partial |\mathbf {x} |{p}}{\partial \mathbf {x} ={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2{|\mathbf {x} |{p}^{p-1}}}.} where {\displaystyle \circ }\circ denotes Hadamard product and {\displaystyle |\cdot |}|\cdot | is used for absolute value of each component of the vector. For the special case of {\displaystyle p=2,}{\displaystyle p=2,} this becomes
{\displaystyle {\frac {\partial }{\partial x_{k}}}|\mathbf {x} |{2}={\frac {x{k}}{|\mathbf {x} |{2}}},}{\displaystyle {\frac {\partial }{\partial x{k}}}|\mathbf {x} |{2}={\frac {x{k}}{|\mathbf {x} |{2}}},}or{\displaystyle {\frac {\partial }{\partial \mathbf {x} }}|\mathbf {x} |{2}={\frac {\mathbf {x} }{|\mathbf {x} |{2}}}.}{\displaystyle {\frac {\partial }{\partial \mathbf {x} }}|\mathbf {x} |{2}={\frac {\mathbf {x} }{|\mathbf {x} |_{2}}}.}
To all our readers in Canada, This Tuesday your support is requested by the nonprofit that hosts Wikipedia and twelve other free knowledge projects. We are financially secure because each year, enough readers choose to donate. Donations support the technology that makes our projects possible, and help us provide resources to the groups who build local communities of contributors to create our millions of articles and images. They also help us advocate for public policy to advance the cause of free knowledge worldwide and defend information access in countries struggling with censorship. Today, we invite you to donate $2 or whatever seems right to you. Show the world that access to independent and unbiased information matters to you. Thank you for your time. — The Wikimedia Foundation How often would you like to donate?
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Yes, I’ll donate $2 No, but maybe later when I have more time Norm (mathematics) From Wikipedia, the free encyclopedia Jump to navigationJump to search This article is about norms of normed vector spaces. For field theory, see Field norm. For ideals, see Ideal norm. For group theory, see Norm (group). For norms in descriptive set theory, see prewellordering. In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.
A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.[1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.
The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".[1] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "{\displaystyle ,\leq ,}{\displaystyle ,\leq ,}" in the homogeneity axiom.[2] It can also refer to a norm that can take infinite values,[3] or to certain functions parametrised by a directed set.[4]
Contents 1 Definition 1.1 Equivalent norms 1.2 Notation 2 Examples 2.1 Absolute-value norm 2.2 Euclidean norm 2.3 Quaternions and octonions 2.4 Finite-dimensional complex normed spaces 2.5 Taxicab norm or Manhattan norm 2.6 p-norm 2.7 Maximum norm (special case of: infinity norm, uniform norm, or supremum norm) 2.8 Zero norm 2.9 Infinite dimensions 2.10 Composite norms 2.11 In abstract algebra 3 Properties 3.1 Equivalence 4 Classification of seminorms: absolutely convex absorbing sets 5 See also 6 References 7 Bibliography Definition Given a vector space {\displaystyle X}X over a subfield {\displaystyle F}F of the complex numbers {\displaystyle \mathbb {C} ,}{\displaystyle \mathbb {C} ,} a norm on {\displaystyle X}X is a real-valued function {\displaystyle p:X\to \mathbb {R} }{\displaystyle p:X\to \mathbb {R} } with the following properties, where {\displaystyle |s|}{\displaystyle |s|} denotes the usual absolute value of a scalar {\displaystyle s}s:[5]
Subadditivity/Triangle inequality: {\displaystyle p(x+y)\leq p(x)+p(y)}{\displaystyle p(x+y)\leq p(x)+p(y)} for all {\displaystyle x,y\in X.}{\displaystyle x,y\in X.} Absolute homogeneity: {\displaystyle p(sx)=\left|s\right|p(x)}{\displaystyle p(sx)=\left|s\right|p(x)} for all {\displaystyle x\in X}x\in X and all scalars {\displaystyle s.}s. Positive definiteness/Point-separating: for all {\displaystyle x\in X,}{\displaystyle x\in X,} if {\displaystyle p(x)=0}p(x) = 0 then {\displaystyle x=0.}x=0. Because property (2.) implies {\displaystyle p(0)=0,}{\displaystyle p(0)=0,} some authors replace property (3.) with the equivalent condition: for every {\displaystyle x\in X,}{\displaystyle x\in X,} {\displaystyle p(x)=0}p(x) = 0 if and only if {\displaystyle x=0.}x=0. A seminorm on {\displaystyle X}X is a function {\displaystyle p:X\to \mathbb {R} }{\displaystyle p:X\to \mathbb {R} } that has properties (1.) and (2.)[6] so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if {\displaystyle p}p is a norm (or more generally, a seminorm) then {\displaystyle p(0)=0}{\displaystyle p(0)=0} and that {\displaystyle p}p also has the following property:
Non-negativity: {\displaystyle p(x)\geq 0}{\displaystyle p(x)\geq 0} for all {\displaystyle x\in X.}{\displaystyle x\in X.} Some authors include non-negativity as part of the definition of "norm", although this is not necessary.
Equivalent norms Suppose that {\displaystyle p}p and {\displaystyle q}q are two norms (or seminorms) on a vector space {\displaystyle X.}X. Then {\displaystyle p}p and {\displaystyle q}q are called equivalent, if there exist two positive real constants {\displaystyle c}c and {\displaystyle C}C with {\displaystyle c>0}c>0 such that for every vector {\displaystyle x\in X,}{\displaystyle x\in X,}
{\displaystyle cq(x)\leq p(x)\leq Cq(x).}{\displaystyle cq(x)\leq p(x)\leq Cq(x).} The relation "{\displaystyle p}p is equivalent to {\displaystyle q}q" is reflexive, symmetric ({\displaystyle cq\leq p\leq Cq}{\displaystyle cq\leq p\leq Cq} implies {\displaystyle {\tfrac {1}{Cp\leq q\leq {\tfrac {1}{cp}{\displaystyle {\tfrac {1}{Cp\leq q\leq {\tfrac {1}{cp}), and transitive and thus defines an equivalence relation on the set of all norms on {\displaystyle X.}X. The norms {\displaystyle p}p and {\displaystyle q}q are equivalent if and only if they induce the same topology on {\displaystyle X.}X.[7] Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.[7] Notation If a norm {\displaystyle p:X\to \mathbb {R} }{\displaystyle p:X\to \mathbb {R} } is given on a vector space {\displaystyle X,}X, then the norm of a vector {\displaystyle z\in X}z\in X is usually denoted by enclosing it within double vertical lines: {\displaystyle |z|=p(z).}{\displaystyle |z|=p(z).} Such notation is also sometimes used if {\displaystyle p}p is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation {\displaystyle |x|}|x| with single vertical lines is also widespread.
Examples Every (real or complex) vector space admits a norm: If {\displaystyle x_{\bullet }=\left(x_{i}\right){i\in I}}{\displaystyle x{\bullet }=\left(x_{i}\right){i\in I}} is a Hamel basis for a vector space {\displaystyle X}X then the real-valued map that sends {\displaystyle x=\sum {i\in I}s{i}x{i}\in X}{\displaystyle x=\sum {i\in I}s{i}x_{i}\in X} (where all but finitely many of the scalars {\displaystyle s_{i}}s_{i} are {\displaystyle 0}{\displaystyle 0}) to {\displaystyle \sum {i\in I}\left|s{i}\right|}{\displaystyle \sum {i\in I}\left|s{i}\right|} is a norm on {\displaystyle X.}X.[8] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.
Absolute-value norm The absolute value
{\displaystyle |x|=|x|}{\displaystyle |x|=|x|} is a norm on the one-dimensional vector spaces formed by the real or complex numbers. Any norm {\displaystyle p}p on a one-dimensional vector space {\displaystyle X}X is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces {\displaystyle f:\mathbb {F} \to X,}{\displaystyle f:\mathbb {F} \to X,} where {\displaystyle \mathbb {F} }\mathbb {F} is either {\displaystyle \mathbb {R} }\mathbb {R} or {\displaystyle \mathbb {C} ,}{\displaystyle \mathbb {C} ,} and norm-preserving means that {\displaystyle |x|=p(f(x)).}{\displaystyle |x|=p(f(x)).} This isomorphism is given by sending {\displaystyle 1\in \mathbb {F} }{\displaystyle 1\in \mathbb {F} } to a vector of norm {\displaystyle 1,}1, which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.
Euclidean norm Further information: Euclidean norm and Euclidean distance On the {\displaystyle n}n-dimensional Euclidean space {\displaystyle \mathbb {R} ^{n},}{\displaystyle \mathbb {R} ^{n},} the intuitive notion of length of the vector {\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)}{\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)} is captured by the formula[9]
{\displaystyle |{\boldsymbol {x}}|{2}:={\sqrt {x{1}^{2}+\cdots +x_{n}^{2}}}.}{\displaystyle |{\boldsymbol {x}}|{2}:={\sqrt {x{1}^{2}+\cdots +x_{n}^{2}}}.} This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem. This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares.[10]
The Euclidean norm is by far the most commonly used norm on {\displaystyle \mathbb {R} ^{n},}{\displaystyle \mathbb {R} ^{n},}[9] 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.
The inner product of two vectors of a Euclidean vector space is the dot product of their coordinate vectors over an orthonormal basis. Hence, the Euclidean norm can be written in a coordinate-free way as
{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.} The Euclidean norm is also called the {\displaystyle L^{2L^{2} norm,[11] {\displaystyle \ell ^{2\ell ^{2} norm, 2-norm, or square norm; see {\displaystyle L^{pL^{p} space. It defines a distance function called the Euclidean length, {\displaystyle L^{2L^{2} distance, or {\displaystyle \ell ^{2}}\ell ^{2} distance.
The set of vectors in {\displaystyle \mathbb {R} ^{n+1}}{\displaystyle \mathbb {R} ^{n+1}} whose Euclidean norm is a given positive constant forms an {\displaystyle n}n-sphere.
Euclidean norm of complex numbers See also: Dot product § Complex vectors The Euclidean norm of a complex number is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane {\displaystyle \mathbb {R} ^{2}.}{\displaystyle \mathbb {R} ^{2}.} This identification of the complex number {\displaystyle x+iy}{\displaystyle x+iy} as a vector in the Euclidean plane, makes the quantity {\textstyle {\sqrt {x^{2}+y^{2}}}}{\textstyle {\sqrt {x^{2}+y^{2}}}} (as first suggested by Euler) the Euclidean norm associated with the complex number.
Quaternions and octonions See also: Quaternion and Octonion There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers {\displaystyle \mathbb {R} ,}{\displaystyle \mathbb {R} ,} the complex numbers {\displaystyle \mathbb {C} ,}{\displaystyle \mathbb {C} ,} the quaternions {\displaystyle \mathbb {H} ,}{\displaystyle \mathbb {H} ,} and lastly the octonions {\displaystyle \mathbb {O} ,}{\displaystyle \mathbb {O} ,} where the dimensions of these spaces over the real numbers are {\displaystyle 1,2,4,{\text{ and }}8,}{\displaystyle 1,2,4,{\text{ and }}8,} respectively. The canonical norms on {\displaystyle \mathbb {R} }\mathbb {R} and {\displaystyle \mathbb {C} }\mathbb{C} are their absolute value functions, as discussed previously.
The canonical norm on {\displaystyle \mathbb {H} }\mathbb {H} of quaternions is defined by
{\displaystyle \lVert q\rVert ={\sqrt {,qq^{}~}}={\sqrt {,q^{}q~={\sqrt {,a^{2}+b^{2}+c^{2}+d^{2}~}{\displaystyle \lVert q\rVert ={\sqrt {,qq^{}~}}={\sqrt {,q^{}q~={\sqrt {,a^{2}+b^{2}+c^{2}+d^{2}~} for every quaternion {\displaystyle q=a+b,\mathbf {i} +c,\mathbf {j} +d,\mathbf {k} }{\displaystyle q=a+b,\mathbf {i} +c,\mathbf {j} +d,\mathbf {k} } in {\displaystyle \mathbb {H} .}{\mathbb {H}}. This is the same as the Euclidean norm on {\displaystyle \mathbb {H} }\mathbb {H} considered as the vector space {\displaystyle \mathbb {R} ^{4}.}{\displaystyle \mathbb {R} ^{4}.} Similarly, the canonical norm on the octonions is just the Euclidean norm on {\displaystyle \mathbb {R} ^{8}.}{\displaystyle \mathbb {R} ^{8}.} Finite-dimensional complex normed spaces On an {\displaystyle n}n-dimensional complex space {\displaystyle \mathbb {C} ^{n},}{\displaystyle \mathbb {C} ^{n},} the most common norm is
{\displaystyle |{\boldsymbol {z}}|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}{1}+\cdots +z{n}{\bar {z}}{n}}}.}{\displaystyle |{\boldsymbol {z}}|:={\sqrt {\left|z{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}{1}+\cdots +z{n}{\bar {z}}_{n}}}.} In this case, the norm can be expressed as the square root of the inner product of the vector and itself:
{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},}{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},} where {\displaystyle {\boldsymbol {x}}}{\boldsymbol {x}} is represented as a column vector {\displaystyle {\begin{bmatrix}x_{1};x_{2};\dots ;x_{n}\end{bmatrix}}^{\rm {T}}}{\displaystyle {\begin{bmatrix}x_{1};x_{2};\dots ;x_{n}\end{bmatrix}}^{\rm {T}}} and {\displaystyle {\boldsymbol {x}}^{H}}{\displaystyle {\boldsymbol {x}}^{H}} denotes its conjugate transpose. This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation:
{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}{\displaystyle |{\boldsymbol {x}}|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.} Taxicab norm or Manhattan norm Main article: Taxicab geometry {\displaystyle |{\boldsymbol {x}}|{1}:=\sum {i=1}^{n}\left|x{i}\right|.}{\displaystyle |{\boldsymbol {x}}|{1}:=\sum {i=1}^{n}\left|x{i}\right|.} The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point {\displaystyle x.}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 {\displaystyle \ell ^{1}}\ell ^{1} norm. The distance derived from this norm is called the Manhattan distance or {\displaystyle \ell _{1}}\ell _{1} distance.
The 1-norm is simply the sum of the absolute values of the columns.
In contrast,
{\displaystyle \sum {i=1}^{n}x{i}}{\displaystyle \sum {i=1}^{n}x{i}} is not a norm because it may yield negative results. p-norm Main article: Lp space Let {\displaystyle p\geq 1}p\geq 1 be a real number. The {\displaystyle p}p-norm (also called {\displaystyle \ell {p}}{\displaystyle \ell {p}}-norm) of vector {\displaystyle \mathbf {x} =(x{1},\ldots ,x{n})}{\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is[9]
{\displaystyle |\mathbf {x} |{p}:=\left(\sum {i=1}^{n}\left|x{i}\right|^{p}\right)^{1/p}.}{\displaystyle |\mathbf {x} |{p}:=\left(\sum {i=1}^{n}\left|x{i}\right|^{p}\right)^{1/p}.} For {\displaystyle p=1,}{\displaystyle p=1,} we get the taxicab norm, for {\displaystyle p=2}p=2 we get the Euclidean norm, and as {\displaystyle p}p approaches {\displaystyle \infty }\infty the {\displaystyle p}p-norm approaches the infinity norm or maximum norm: {\displaystyle |\mathbf {x} |{\infty }:=\max {i}\left|x{i}\right|.}{\displaystyle |\mathbf {x} |{\infty }:=\max {i}\left|x{i}\right|.} The {\displaystyle p}p-norm is related to the generalized mean or power mean. For {\displaystyle p=2,}{\displaystyle p=2,} the {\displaystyle |,\cdot ,|{2}}{\displaystyle |,\cdot ,|{2}}-norm is even induced by a canonical inner product {\displaystyle \langle ,\cdot ,,\cdot \rangle ,}{\displaystyle \langle ,\cdot ,,\cdot \rangle ,} meaning that {\displaystyle |\mathbf {x} |{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}}{\displaystyle |\mathbf {x} |{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} for all vectors {\displaystyle \mathbf {x} .}{\displaystyle \mathbf {x} .} This inner product can expressed in terms of the norm by using the polarization identity. On {\displaystyle \ell ^{2},}{\displaystyle \ell ^{2},} this inner product is the Euclidean inner product defined by
{\displaystyle \langle \left(x_{n}\right){n},\left(y{n}\right){n}\rangle {\ell ^{2}}~=~\sum {n}{\overline {x{n}y{n{\displaystyle \langle \left(x{n}\right){n},\left(y{n}\right)_{n}\rangle {\ell ^{2}}~=~\sum {n}{\overline {x{n}y{n while for the space {\displaystyle L^{2}(X,\mu )}L^{2}(X,\mu ) associated with a measure space {\displaystyle (X,\Sigma ,\mu ),}{\displaystyle (X,\Sigma ,\mu ),} which consists of all square-integrable functions, this inner product is {\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)g(x),\mathrm {d} x.}{\displaystyle \langle f,g\rangle _{L^{2=\int _{X}{\overline {f(x)g(x),\mathrm {d} x.}This definition is still of some interest for {\displaystyle 0<p<1,}{\displaystyle 0<p<1,} but the resulting function does not define a norm,[12] because it violates the triangle inequality. What is true for this case of {\displaystyle 0<p<1,}{\displaystyle 0<p<1,} even in the measurable analog, is that the corresponding {\displaystyle L^{pL^{p} class is a vector space, and it is also true that the function
{\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu }{\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu } (without {\displaystyle p}pth root) defines a distance that makes {\displaystyle L^{p}(X)}{\displaystyle L^{p}(X)} into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic analysis. However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional. The partial derivative of the {\displaystyle p}p-norm is given by
{\displaystyle {\frac {\partial }{\partial x_{k}}}|\mathbf {x} |{p}={\frac {x{k}\left|x_{k}\right|^{p-2}}{|\mathbf {x} |{p}^{p-1}}}.}{\displaystyle {\frac {\partial }{\partial x{k}}}|\mathbf {x} |{p}={\frac {x{k}\left|x_{k}\right|^{p-2}}{|\mathbf {x} |_{p}^{p-1}}}.} The derivative with respect to {\displaystyle x,}x, therefore, is
{\displaystyle {\frac {\partial |\mathbf {x} |{p}}{\partial \mathbf {x} ={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2{|\mathbf {x} |{p}^{p-1}}}.}{\displaystyle {\frac {\partial |\mathbf {x} |{p}}{\partial \mathbf {x} ={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2{|\mathbf {x} |{p}^{p-1}}}.} where {\displaystyle \circ }\circ denotes Hadamard product and {\displaystyle |\cdot |}|\cdot | is used for absolute value of each component of the vector. For the special case of {\displaystyle p=2,}{\displaystyle p=2,} this becomes
{\displaystyle {\frac {\partial }{\partial x_{k}}}|\mathbf {x} |{2}={\frac {x{k}}{|\mathbf {x} |{2}}},}{\displaystyle {\frac {\partial }{\partial x{k}}}|\mathbf {x} |{2}={\frac {x{k}}{|\mathbf {x} |{2}}},}or{\displaystyle {\frac {\partial }{\partial \mathbf {x} }}|\mathbf {x} |{2}={\frac {\mathbf {x} }{|\mathbf {x} |{2}}}.}{\displaystyle {\frac {\partial }{\partial \mathbf {x} }}|\mathbf {x} |{2}={\frac {\mathbf {x} }{|\mathbf {x} |_{2}}}.}
