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Conversely, any vector space,, is an affine space over itself.

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## Some Related Sentences

Conversely and any

__Conversely__

**,**a subset R defines a binary function if and only if

**,**for

__any__x in X and y in Y

**,**there exists a unique z in Z such that ( x

**,**y

**,**z ) belongs to R

**.**

__Conversely__the equilibrium position

**is**said to be " far to the left " if hardly

__any__product

**is**formed from the reactants

**.**

* The spectrum of

__any__bounded linear operator on a Banach**space****is**a nonempty compact subset of the complex numbers C**.**__Conversely__**,**__any__compact subset of C arises in this manner**,**as the spectrum of some bounded linear operator**.**__Conversely__

**,**there are other stars that never rise above the horizon

**,**as seen from

__any__given point on the Earth's surface ( except exactly on the equator ).

__Conversely__

**,**

__any__surjection between sets determines a partition on its domain

**,**the set of preimages of singletons in the codomain

**.**

__Conversely__

**,**Derek Bickerton and others argue that the advent of abstract words provided a mental basis for analyzing higher-order relations

**,**and that

__any__communication system that remotely resembles human language utterly relies on cognitive architecture that co-evolved alongside language

**.**

__Conversely__

**,**a person may feel like he belongs to one state without having

__any__legal relationship to it

**.**

__Conversely__

**,**the interior of the box has a constant

**,**zero potential energy .< ref > Actually

**,**

__any__constant

**,**finite potential can be specified within the box

**.**

__Conversely__

**,**extremely thin jewelry can cause the same tearing in what

**is**commonly referred to as the " cheese cutter effect ", either during sudden torsion or

**over**a long period of wearing

**,**especially if the thin jewelry bears

__any__weight

**.**

__Conversely__

**,**

__any__Hermitian positive semidefinite matrix M can be written as M = A < sup >*</ sup > A ; this

**is**the Cholesky decomposition

**.**

__Conversely__

**,**if a double sharp were added to

__any__other note not containing a sharp or flat as indicated by the key signature then the note will be raised two semitones or a whole step with respect to the chromatic scale

**.**

__Conversely__

**,**supplementation with L-phenylalanine

**,**L-tyrosine

**,**L-DOPA

**,**or

__any__of the cofactors will increase their respective concentrations

**.**

__Conversely__

**,**

__any__choice of μ

**,**full rank matrix U

**,**and positive diagonal entries Λ < sub > i </ sub > yields a non-singular multivariate normal distribution

**.**

__Conversely__

**,**the influence of the data at

__any__given point on the initial line propagates with the finite velocity c: there

**is**no effect outside a triangle through that point whose sides are characteristic curves

**.**

__Conversely__

**,**given

__any__harmonic function in two dimensions

**,**it

**is**the real part of

**an**analytic function

**,**at least locally

**.**

__Conversely__

**,**if the Turing Machine

**is**expected polynomial-time ( for

__any__given x ), then a considerable fraction of the runs must be polynomial-time bounded

**,**and the coin sequence used in such a run will be a witness

**.**

It

**is**a fact that the ring**is**a principal ideal ring ; that**is****,**for__any__ideal I in**,**there exists**an**integer n in I such that every element of I**is**a multiple of n**.**__Conversely__**,**the set of all multiples of**an**arbitrary integer n**is**necessarily**an**ideal**,**and**is**usually denoted by ( n ).

Conversely and vector

__Conversely__

**,**one can consider the flux the more fundamental quantity and call the

__vector__field the flux density

**.**

Given a

__vector__**space**V**over**a field K**,**the span of a set S ( not necessarily finite )**is**defined to be the intersection W of all subspaces of V which contain S**.**W**is**referred to as the subspace spanned by S**,**or by the vectors in S**.**__Conversely__**,**S**is**called a spanning set of W**.**__Conversely__

**,**a system of n quantities v < sup > i </ sup > that transform like the coordinates x < sup > i </ sup > on V defines a contravariant

__vector__

**.**

for some real-valued functions a and b depending

**,**respectively**,**on x and y**.**__Conversely__**,**given**any**such pair of real-valued functions**,**there exists a__vector__field X satisfying 1**.**and 2**.**__Conversely__the index theorem for

**an**elliptic complex can easily be reduced to the case of

**an**elliptic operator: the two

__vector__bundles are given by the sums of the even or odd terms of the complex

**,**and the elliptic operator

**is**the sum of the operators of the elliptic complex and their adjoints

**,**restricted to the sum of the even bundles

**.**

__Conversely__

**,**if a highest weight module

**is**integrable

**,**then its highest weight

__vector__v satisfies

**,**where are complex numbers such that

A section Γ of dπ: TP / G → TM such that Γ: TM → TP / G

**is**a linear morphism of__vector__bundles**over**M**,**can be identified with a principal connection in P**.**__Conversely__**,**a principal connection as defined above gives rise to such a section Γ of TP / G**.**__Conversely__

**,**given a left kG module M

**,**then M

**is**a k

__vector__

**space**

**,**and multiplication with

**an**element g of G yields a k-linear automorphism of M ( since g

**is**invertible in kG ), which describes a group homomorphism G → GL ( M ).

__Conversely__

**,**MAC algorithms are designed to produce matching MACs only if the same message

**,**secret key and initialization

__vector__are input to the same algorithm

**.**

__Conversely__

**,**if the holonomy group of the Riemannian manifold M

**is**contained in Sp ( k ), choose complex structures I < sub > x </ sub >, J < sub > x </ sub > and K < sub > x </ sub > on T < sub > x </ sub > M which make T < sub > x </ sub > M into a quaternionic

__vector__

**space**

**.**

Conversely and space

__Conversely__

**,**because

__space__

**is**expanding

**,**and more distant objects are receding ever more quickly

**,**light emitted by us today may never " catch up " to very distant objects

**.**

__Conversely__

**,**one might expect that inertial motions

**,**once identified by observing the actual motions of bodies and making allowances for the external forces ( such as electromagnetism or friction ), can be used to define the geometry of

__space__

**,**as well as a time coordinate

**.**

__Conversely__

**,**in the bleak Orwellian world of " The High Ones "— where the Soviets have won the Third World War and gained control of the whole world — the dissidents still have some hope

**,**precisely because

__space__flight has not been abandoned

**.**

__Conversely__

**,**a metrizable

__space__

**is**separable if and only if it

**is**second countable

**,**which

**is**the case if and only if it

**is**Lindelöf

**.**

__Conversely__

**,**except in dimension 2

**,**if the curvature of a Riemannian manifold has this form for some function K

**,**then the Bianchi identities imply that K

**is**constant and thus that the manifold

**is**( locally ) a

__space__form

**.**

Equipped with this notion of length

**,**a Riemannian manifold**is**a metric__space__**,**meaning that it has a distance function d ( p**,**q ) whose value at a pair of points p and q**is**the distance from p to q**.**__Conversely__**,**the metric tensor**itself****is**the derivative of the distance function ( taken in a suitable manner ).__Conversely__

**,**every Polish

__space__

**is**homeomorphic to a G < sub > δ </ sub >- subset of the Hilbert cube

**.**

__Conversely__

**,**when a record

**is**deleted

**,**later records are moved down so that the free

__space__remains contiguous

**.**

__Conversely__

**,**if X

**is**projective

**,**then the pullback of under the closed immersion of X into a projective

__space__

**is**very ample

**.**

__Conversely__

**,**if X

**is**a Hausdorff

__space__and ker f

**is**a closed set

**,**then the coimage of f

**,**if given the quotient

__space__topology

**,**must also be a Hausdorff

__space__

**.**

Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological

__space__1 ( all subsets of which are open ) to the__space__X by defining p < sub > x </ sub >( 1 ) = x**.**__Conversely__**,****any**function from 1 to X clearly determines one point: the element that it " points " to**.**__Conversely__

**,**

**any**such one form defines ( via pullback ) a G-equivariant horizontal 1-form on P

**,**and the

__space__of principal G-connections

**is**

**an**

**affine**

__space__for this

__space__of 1-forms

**.**

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