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Formally, an inner product space is a vector space V over the field together with an inner product, i. e., with a map

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## Some Related Sentences

Formally and inner

Formally and product

__Formally__

**,**we start

**with**

**a**category C

**with**finite products (

**i**

**.**

**e**

**.**C has

**a**terminal object 1 and any two objects of C have

**a**

__product__).

__Formally__

**,**

**a**unique factorization domain

**is**defined to be

**an**integral domain R in which every non-zero and non-unit x of R can be written as

**a**

__product__( including

**an**empty

__product__) of irreducible elements p < sub >

**i**</ sub > of R and

**a**unit u:

__Formally__

**,**two variables are inversely proportional ( or varying inversely

**,**or in inverse variation

**,**or in inverse proportion or in reciprocal proportion ) if one of

**the**variables

**is**directly proportional

**with**

**the**multiplicative inverse ( reciprocal ) of

**the**other

**,**or equivalently if their

__product__

**is**

**a**constant

**.**

__Formally__

**,**

**a**Lie superalgebra

**is**

**a**( nonassociative ) Z < sub > 2 </ sub >- graded algebra

**,**or superalgebra

**,**

**over**

**a**commutative ring ( typically R or C ) whose

__product__

**,**called

**the**Lie superbracket or supercommutator

**,**satisfies

**the**two conditions ( analogs of

**the**usual Lie algebra axioms

**,**

**with**grading ):

__Formally__

**,**

**the**differential appearing under

**the**integral behaves exactly as

**a**differential: thus

**,**

**the**integration by substitution and integration by parts formulae for Stieltjes integral correspond

**,**respectively

**,**to

**the**chain rule and

__product__rule for

**the**differential

**.**

__Formally__

**,**f < sub > X

**,**Y </ sub >( x

**,**y )

**is**

**the**probability density function of ( X

**,**Y )

**with**respect to

**the**

__product__measure on

**the**respective supports of X and Y

**.**

__Formally__

**,**

**a**

__product__term P in

**a**sum of products

**is**

**an**implicant of

**the**Boolean function F if P implies F

**.**More precisely:

Formally and space

__Formally__

**,**

**a**profinite group

**is**

**a**Hausdorff

**,**compact

**,**and totally disconnected topological group: that

**is**

**,**

**a**topological group that

**is**also

**a**Stone

__space__

**.**

__Formally__

**,**these reside in

**a**complex separable Hilbert space-variously called

**the**" state

__space__" or

**the**" associated Hilbert

__space__" of

**the**system-that

**is**well defined up to

**a**complex number of norm 1 (

**the**phase factor ).

__Formally__

**,**this means symmetry under

**a**sub-group of

**the**Euclidean group of isometries in two or three dimensional Euclidean

__space__

**.**

__Formally__

**,**

**the**question of whether

**the**universe

**is**infinite or finite

**is**whether it

**is**

**an**unbounded or bounded metric

__space__

**.**

__Formally__

**,**Minkowski

__space__

**is**

**a**four-dimensional real

**vector**

__space__equipped

**with**

**a**nondegenerate

**,**symmetric bilinear form

**with**signature < tt >(−,+,+,+)</ tt > ( Some may also prefer

**the**alternative signature < tt >(+,−,−,−)</ tt >; in general

**,**mathematicians and general relativists prefer

**the**former while particle physicists tend to use

**the**latter

**.**

__Formally__

**,**it

**is**

**a**norm defined on

**the**

__space__of bounded linear operators between two given normed

**vector**spaces

**.**

__Formally__

**,**

**a**ringed

__space__( X

**,**O < sub > X </ sub >)

**is**

**a**topological

__space__X

**together**

**with**

**a**sheaf of rings O < sub > X </ sub > on X

**.**

__Formally__

**,**

**a**coalgebra

**over**

**a**

**field**K

**is**

**a**

**vector**

__space__C

**over**K

**together**

**with**K-linear maps Δ: C → C ⊗ C and ε: C → K such that

__Formally__

**,**

**an**ultrametric

__space__

**is**

**a**set of points

**with**

**an**associated distance function ( also called

**a**metric )

__Formally__

**,**

**a**rigged Hilbert

__space__consists of

**a**Hilbert

__space__H

**,**

**together**

**with**

**a**subspace Φ which carries

**a**finer topology

**,**that

**is**one for which

**the**natural inclusion

__Formally__

**,**

**a**frame on

**a**homogeneous

__space__G / H consists of

**a**point in

**the**tautological bundle G → G / H

**.**

__Formally__

**,**

**an**iterated function system

**is**

**a**finite set of contraction mappings on

**a**complete metric

__space__

**.**

__Formally__

**,**rotational symmetry

**is**symmetry

**with**respect to some or all rotations in m-dimensional Euclidean

__space__

**.**

__Formally__

**,**

**a**complex projective

__space__

**is**

**the**

__space__of complex lines through

**the**origin of

**an**( n + 1 )- dimensional complex

**vector**

__space__

**.**

Formally and is

__Formally__organized vocational programs supported by federal funds allow high school students to gain experience in

**a**

**field**of work which

__is__likely to lead to

**a**full-time job on graduation

**.**

__Formally__

**,**

**the**set of all context-free languages

__is__identical to

**the**set of languages accepted by pushdown automata ( PDA ).

More rigorously

**,****the**divergence of**a****vector****field**F at**a**point p__is__defined as**the**limit of**the**net flow of F across**the**smooth boundary of**a**three dimensional region**V**divided by**the**volume of**V**as**V**shrinks to p**.**__Formally__**,**__Formally__

**,**

**the**base

__is__known as Naval Support Facility Diego Garcia (

**the**US activity ) or Permanent Joint Operating Base ( PJOB ) Diego Garcia (

**the**UK's term ).

__Formally__

**,**there

__is__

**a**clear distinction: " DFT " refers to

**a**mathematical transformation or function

**,**regardless of how it

__is__computed

**,**whereas " FFT " refers to

**a**specific family of algorithms for computing DFTs

**.**

__Formally__

**,**oxidation state

__is__

**the**hypothetical charge that

**an**atom would have if all bonds to atoms of different elements were 100 % ionic

**.**

__Formally__

**,**if M

__is__

**a**set

**,**

**the**identity function f on M

__is__defined to be that function

**with**domain and codomain M which satisfies

*

__Formally__**,**when working**over****the**reals**,**as here**,**this__is__accomplished by considering**the**limit as ε → 0 ; but**the**" infinitesimal " language generalizes directly to Lie groups**over**general rings**.**__Formally__

**,**this sharing of dynamics

__is__referred to as universality

**,**and systems

**with**precisely

**the**same critical exponents are said to belong to

**the**same universality class

**.**

__Formally__

**,**

**a**frame

__is__defined to be

**a**lattice L in which finite meets distribute

**over**arbitrary joins

**,**

**i**

**.**

**e**

**.**every ( even infinite ) subset

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