Normalizes (mathematical)

Usual Euclidean geometry

Definition

If has and B is two points of the plan or usual space, the standard of the vector $\ overrightarrow {AB}$ is the distance $AB$ i.e. the length of the segment . It is noted using a double bar: $\|\overrightarrow{AB}\|$ .

The standard, the direction and the direction are the three data which characterize a vector and which thus do not depend on the choice of the representative.

Calculation

The standard of a vector can be calculated using its coordinates in a Repère orthonormé using the theorem of Pythagore.

In the plan, if the vector $\ vec u$ has as coordinates $\ (X; there)$ its standard is written $\|\ vec U \| = \ sqrt {x^2 + y^2}$ .

If the points has and B has as respective coordinates

\ (x_A, y_A)

and

\ (x_B, y_B)

then

\|\overrightarrow{AB}\| = \ sqrt {(x_B-x_A) ^2 + (y_B there _A) ^2}

In space, if the vector $\ vec u$ has as coordinates $\ (X; there; Z)$ its standard is written $\|\ vec U \| = \ sqrt {x^2 + y^2 + z^2}$ .

If the points has and B has as respective coordinates

\ (x_A; y_A; z_A)

and

\ (x_B; y_B; z_B)

then

\|\overrightarrow{AB}\| = \ sqrt {(x_B-x_A) ^2 + (y_B there _A) ^2 + (z_B-z_A) ^2}

The standard of a vector can be obtained starting from the scalar Produit:

\|\ vec U \| = \ sqrt {\ vec U \ cdot \ vec U}

Reciprocally, the scalar product can be obtained on the standard basis thanks to the relation:

\ vec U \ cdot \ vec v = \ frac {1} {2} \ left (\|\ vec U + \ vec v \|^2 - \|\ vec U \|^2 - \|\ vec v \|^2 \ right)

Properties

the standard cancels only for the null vector $\ vec 0$ .
the standard of the product by a number is the product of the standard by the absolute value of this number:

\|K. \ vec U \| = |K|\ times \|\ vec U \|

In particular, any vector to the same standard as its opposite:

\|- \ vec U \| = \|\ vec U \|

On an unspecified vector space

Formal definition

That is to say K a body provided with a absolute value and E a K - vector Space.

A standard on E is an application $\ mathcal N$ on E to real values positive and satisfying the following assumptions:

separation : $\ forall X \ in E, \ \ mathcal NR (X) =0 \ Rightarrow x=0_E$ ;
homogeneity : $\ forall (\ lambda, X) \ in \ mathbb K \ times E, \ \ mathcal NR (\ lambda \ cdot X) = |\ lambda| \ mathcal NR (X)$ ;
under-additivity : $\ forall (X, there) \ in E^2, \ \ mathcal NR (X + there) \ Leq \ mathcal NR (X) + \ mathcal NR (there)$ .

; Note:

the bodies of realities and the complexes are not the only ones to admit an absolute value. Any finished body supports the constant absolute value equal to 1 apart from 0.
Danslecasde the Corps valué S, the standard is even ultrametric by checking a certain condition stronger than under-additivity.
a function of E in $\ R^+$ which satisfies only the assumptions of homogeneity and under-additivity is called Semi-norme.

A vector space provided with a standard is then called vector space normalized (sometimes shortened in EVN).

First properties

the standard is under-linear , i.e. it checks the following property:

\ forall (\ lambda, X, there) \ in \ mathbb K \ times E^2 \, \ \|(\ lambda \ cdot X + there) \| \ Leq |\ lambda|\ cdot \|X \| + \|there \|

more generally, one obtains by immediate recurrence the inequality in $\ R$ :

\|\ lambda_1 \ cdot x_1 + \ dowries + \ lambda_n \ cdot x_n \| \ Leq | \ lambda_1|\ times \|x_1 \|+ \ dowries + |\ lambda_n|\ times \|x_n \| \ \

separation and the homogeneity guarantee the properties of separation and symmetry of the function $d \ colonist (X, there) \ mapsto \|y-x \|$ . Under-additivity justifies the triangular Inégalité then,

\|Z - X \| \ Leq \|Z - there \| + \|there - X \|

necessary to show that D is a distance on E , which more is invariant by translation.

a normalized vector space is thus a metric Espace homogeneous and the associated topology is compatible with the vectorial operations.

under-additivity makes it possible to obtain the following property:

\ forall (X, there) \ in E^2 \, \ \ big| \|X \|- \|there \|\ big| \ Leq \|X there \|

which shows that the standard is an application 1-lipschitzienne thus continuous.

the standard is also a convex Fonction, which can be useful to solve problems of optimization.

Topology

The Boule unit (open) $\ mathcal {B}$ of a standard is the whole of the vectors of standard lower (strictly) than 1.

Two standards $\ mathcal N_1$ and $\ mathcal N_2$ on a vector space E are known as equivalent if there exist two strictly positive realities $\ alpha$ and $\ beta$ such as:

\ forall X \ in E, \ \ alpha N_1 (X) \ Leq N_2 (X) \ Leq \ beta N_1 (X)

That corresponds to the fact that in the open balls of the two standards can include one in the other except for dilation.

Two equivalent standards define same topology on the vector space. The structures are even uniformly isomorphous.

On a real or complex vector space of finished size, all the standards are equivalent.

The ball open unit is an open convex limited and balanced of E .

Generic constructions

All Produit scalar on a real vector space E defines the euclidian norm associated by:

\ forall X \ in E, \ \|X \| = \ sqrt {\ langle X, X \ rangle}

a standard

\ mathcal N

is Euclidean (i.e. comes from a scalar product) if and only if the application

(X, there) \ mapsto \ frac {1} {2} (\ mathcal NR (x+y) ^2- \ mathcal NR (X) ^2 - \ mathcal NR (there) ^2)

is Bilinéaire

and in this case this application is the associated scalar product.

If F is an injective linear application of E in F then any standard on F induces a standard on E by the equation

\ mathcal \|X \|_E = \|F (X) \|_F

If C is an open convex limited and balanced of a real or complex vector space E , then the gauge of C is a standard defined by

\ forall X \ in E \, \ J (X) = \ inf \ left \ {\ lambda \ in \ R^+ \ colonist \ frac {1} {\ lambda} X \ in C \ right \}

and whose C is the Boule open unit.

If E and F is two real or complex normalized vector spaces, space $\ L_c (E, F)$ of the continuous linear applications is provided with the Norme of being written operator :

\ forall T \ in L_c (E, F), \ \|T \| = \ sup_ {X \ in E \ {0 \}} \ frac {\|T (X) \|_F} {\|X \|_E}

Examples

In finished dimension

On K ^N,

the euclidian norm is obtained starting from the scalar Produit or of the square produces canonical:

\|(x_1, \ dowries, x_n) \| = \ sqrt

In infinite dimension

On space $\ mathcal C^0 ()$ of the continuous functions definite on a segment of $\ R$ and with actual values or complex, one finds standards p definite in ways similar to those on the vector spaces of size finished for p equal to or higher than 1:

{\|F \|} _p = \ left (\ int_a^b |F (T)|^p \ mathrm dt \ right) ^ {1/p}

which in particular makes it possible to define the spaces '' L^p ''.

In particular, the euclidian norm associated with the scalar or square product canonical is defined by

\|F \| = \ sqrt {\ int_a^b |F (T)|^2\mathrm dt}

the standard infinite or standard sup or standard of the uniform Convergence is written as for it

{\|F \|} _ {\ infty} = \ sup_ {T \ in B} |F (T)|

and is obtained there too as limit of the standards p when p tends towards the infinite one.

All these standards are not equivalent two to two.

In addition they easily extend to spaces from continuous functions on a compact from

\ R^n

, even with the continuous functions with compact support.

the continuity of the functions also makes it possible to define more exotic standards as this one:

(u_n) _ {N \ in \ NR}

is a dense continuation of image in the field of definition,

\ mathcal NR (F) = \ sum_n \ frac {1} {2^ {n+1}}|F (u_n)|

. -->

On space $\ mathcal C^1 ()$ of the derivable functions with continuous derivative, one can use one of the standards above or as follows take into account also the derivative using a standard:

\|F \| = \ int_a^b (|F (T)| + |f' (T)|\ mathrm dt)

in order to regard the application derived from

\ mathcal C^1 ()

\ mathcal C^0 ()

as continuous.

On space $\ ell^ {\ infty}$ of the limited continuations, the natural standard is the standard sup :

{\|(u_n) _ {N \ in \ NR} \|} _ {\ infty} = \ sup_ {N \ in \ NR}|u_n|

but other standards can be written in the following forms:

(u_n) _ {N \ in \ NR} \ mapsto \ sup_ {N \ in \ NR}|\ cos (N) \ times u_n|\ quad

\ quad (u_n) _ {N \ in \ NR} \ mapsto \ sum_ {N \ in \ NR} \ frac {1 + n^2}

. -->

Algebra normalizes

Definition

A standard $\ mathcal N$ on an algebra has is known as standard of algebra if there exists a real constant C such as

\ forall (X, there) \ in A^2, \ mathcal NR (X \ times there) \ Leq \ mathcal C NR (X) \ times \ mathcal NR (there)

Even if it means to multiply the standard by C , this constant can be brought back to 1. The condition is then that of under-multiplicativité.

In the case of a real or complex algebra, the condition is equivalent to the continuity of the product like bilinear application.

If the algebra is unit, one can require standard which it also checks:

\ mathcal NR (1_A) =1

in which case the multiplication by a constant cannot be any more used for “renormaliser” the standard.

Examples

the application module is a standard of algebra on $\ mathbb {C}$ considered as $\ R$ -algèbre.
the standard of operator on $\ L_c (E)$ is a standard of algebra.
the standard infinite on $\ mathbb {C} ^n$ induces the standard of operator on $\ \ mathbb \ mathcal M_n (\ mathbb C)$ which is written

\ forall (a_ {I, J}) \ in \ mathcal {M} _n (\ mathbb C), \ \|(a_ {ij}) \| = \ max_i \ sum_j|a_ {ij}|

but the standard defined by

\ forall (a_ {I, J}) \ in \ mathcal {M} _n (\ mathbb C), \ \|(a_ {ij}) \|_ \ infty = \ max_ {I, J}|a_ {I, J}|

is not a standard of algebra because it is not under-multiplicative as soon as N is larger than 2. -->

\ mathcal A

is a limited infinite part of

\ R

(or

\ mathbb C

), the algebra

\ mathbb K

of the polynomials on

\ mathbb K

can be normalized by the standard of following algebra:

\ forall P \ in \ mathbb K, \ \| P \|_ \ infty^ {\ mathcal has} = \ sup_ {X \ in \ mathcal has}|P (X)|

TO MOVE IN a PAGE Of HOMONYMY

Normalizes in a body of numbers

Zero normalize

If $\ mathbb K$ is a body, the standard zero of a vector X of $\ mathbb K^n$ is the number of nonnull coordinates of X .
It is not a question of a standard within the meaning of the linear algebra since it is neither homogeneous nor under-additive, but its name comes from the writing

\|X \|_0 = \ lim_ {p \ rightarrow 0} {\|X \|_p} ^p

when

\ mathbb K = \ mathbb R

where the “standards p ” when p lies between 0 and 1, are defined in manner analogues to the true standards p .

In the context of the error correcting codes, the standard zero is also called standard of Hamming with $\ mathbb K = \ mathbb Z/2$ . -->

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