[permalink] [id link]

Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).

from
Wikipedia

## Some Related Sentences

Every and finite-dimensional

*

__Every__commutative real unital Noetherian Banach algebra**(**possibly having zero divisors )**is**__finite-dimensional__.__Every__module over

**a**division ring has

**a**

**basis**; linear maps between

__finite-dimensional__modules over

**a**division ring can be described by matrices

**,**

**and**

**the**Gaussian elimination algorithm remains applicable.

__Every__

__finite-dimensional__inner product

**space**has

**an**orthonormal

**basis**

**,**which may be obtained from

**an**

**arbitrary**

**basis**using

**the**Gram – Schmidt process.

__Every__

__finite-dimensional__normed

**space**

**is**reflexive

**,**simply because in

**this**case

**,**

**the**

**space**

**,**

**its**

**dual**

**and**bidual all have

**the**same linear dimension

**,**hence

**the**linear injection J from

**the**definition

**is**bijective

**,**by

**the**rank-nullity theorem.

__Every__

__finite-dimensional__Hausdorff topological

**vector**

**space**

**is**reflexive

**,**because J

**is**bijective by linear algebra

**,**

**and**because there

**is**

**a**unique Hausdorff

**vector**

**space**topology

**on**

**a**finite dimensional

**vector**

**space**.

*

__Every____finite-dimensional__simple algebra over R must be**a**matrix ring over R**,**C**,**or H.__Every__central simple algebra over R must be**a**matrix ring over R or H. These results follow from**the**Frobenius theorem.
*

__Every____finite-dimensional__simple algebra over C must be**a**matrix ring over C**and**hence every central simple algebra over C must be**a**matrix ring over C.
*

__Every____finite-dimensional__central simple algebra over**a**finite field must be**a**matrix ring over that field.

Every and vector

__Every__subset A

**of**

**the**

__vector__

**space**

**is**contained within

**a**smallest convex set

**(**called

**the**convex hull

**of**A ), namely

**the**intersection

**of**all convex sets containing A.

__Every__

__vector__v in determines

**a**linear map from R

**to**

**taking**1

**to**v

**,**which can be thought

**of**as

**a**Lie algebra homomorphism.

__Every__

__vector__

**space**has

**a**

**basis**

**,**

**and**all bases

**of**

**a**

__vector__

**space**have

**the**same number

**of**elements

**,**called

**the**dimension

**of**

**the**

__vector__

**space**.

__Every__normed

__vector__

**space**V sits as

**a**dense subspace inside

**a**Banach

**space**;

**this**Banach

**space**

**is**essentially uniquely defined by V

**and**

**is**called

**the**completion

**of**V.

__Every__random

__vector__gives rise

**to**

**a**probability measure

**on**R < sup > n </ sup > with

**the**Borel algebra as

**the**underlying sigma-algebra.

__Every__continuous function in

**the**function

**space**can be represented as

**a**linear combination

**of**

**basis**functions

**,**just as every

__vector__in

**a**

__vector__

**space**can be represented as

**a**linear combination

**of**

**basis**vectors.

__Every__coalgebra

**,**by

**(**

__vector__

**space**) duality

**,**gives rise

**to**

**an**algebra

**,**

**but**not in general

**the**other way.

*

__Every__holomorphic__vector__bundle**on****a**projective variety**is**induced by**a**unique algebraic__vector__bundle.
For

**example****,**second-order arithmetic can express**the**principle "__Every__countable__vector__**space**has**a****basis**"**but**it cannot express**the**principle "__Every____vector__**space**has**a****basis**".
Many principles that imply

**the**axiom**of****choice**in their general form**(**such as "__Every____vector__**space**has**a****basis**") become provable in weak subsystems**of**second-order arithmetic when they are restricted.__Every__

__vector__

**space**

**is**free

**,**

**and**

**the**free

__vector__

**space**

**on**

**a**set

**is**

**a**special case

**of**

**a**free module

**on**

**a**set.

Every and space

__Every__Hilbert

__space__X

**is**

**a**Banach

__space__because

**,**by definition

**,**

**a**Hilbert

__space__

**is**complete with respect

**to**

**the**norm associated with

**its**inner product

**,**where

**a**norm

**and**

**an**inner product are said

**to**be associated if

**for**all x ∈ X.

*

__Every__topological__space__X**is****a**dense subspace**of****a**compact__space__having at most one point more than X**,**by**the**Alexandroff one-point compactification.
*

__Every__continuous map from**a**compact__space__**to****a**Hausdorff__space__**is**closed**and**proper**(**i. e.,**the**pre-image**of****a**compact set**is**compact.__Every__node

**on**

**the**Freenet network contributes storage

__space__

**to**hold files

**,**

**and**bandwidth that it uses

**to**route requests from

**its**peers.

*

__Every__Lie group**is**parallelizable**,****and**hence**an**orientable manifold**(**there**is****a**bundle**isomorphism**between**its**tangent bundle**and****the**product**of**itself with**the**tangent__space__at**the**identity )

Every and is

__Every__legislator from Brasstown Bald

**to**Folkston

__is__going

**to**have his every vote subjected

**to**

**the**closest scrutiny as

**a**test

**of**his political allegiances

**,**not his convictions.

__Every__detail in his interpretation has been beautifully thought out

**,**

**and**

**of**these I would especially cite

**the**delicious laendler touch

**the**pianist brings

**to**

**the**fifth variation

**(**

**an**obvious indication that he

__is__playing with Viennese musicians )

**,**

**and**

**the**gossamer shading throughout.

__Every__taxpayer

__is__well aware

**of**

**the**vast size

**of**our annual defense budget

**and**most

**of**our readers also realize that

**a**large portion

**of**these expenditures go

**for**military electronics.

__Every__few days

**,**in

**the**early morning

**,**as

**the**work progressed

**,**twenty men would appear

**to**push it ahead

**and**

**to**shift

**the**plank foundation that distributed

**its**weight widely

**on**

**the**Rotunda pavement

**,**supported as it

__is__by ancient brick vaulting.

__Every__dream

**,**

**and**

**this**

__is__true

**of**

**a**mental image

**of**any type even though it may be readily interpreted into

**its**equivalent

**of**wakeful thought

**,**

__is__

**a**psychic phenomenon

**for**which no explanation

__is__available.

__Every__library borrower

**,**or at least those whose taste goes beyond

**the**five-cent fiction rentals

**,**knows what it

__is__

**to**hear

**the**librarian say apologetically

**,**`` I'm sorry

**,**

**but**we don't have that book.

__Every__community

**,**if it

__is__alive has

**a**spirit

**,**

**and**that spirit

__is__

**the**center

**of**

**its**unity

**and**identity.

The restricted principle "

__Every__partially ordered set has**a**maximal totally ordered subset "__is__also equivalent**to**AC over ZF.__Every__natural-born citizen

**of**

**a**foreign state who

__is__also

**an**American citizen

**and**every natural-born American citizen who

__is__

**a**citizen

**of**

**a**foreign land owes

**a**double allegiance

**,**one

**to**

**the**United States

**,**

**and**one

**to**his homeland

**(**in

**the**event

**of**

**an**immigrant becoming

**a**citizen

**of**

**the**US ), or

**to**his adopted land

**(**in

**the**event

**of**

**an**emigrant natural born citizen

**of**

**the**US becoming

**a**citizen

**of**another nation

**).**

__Every__line

**of**written text

__is__

**a**mere reflection

**of**references from any

**of**

**a**multitude

**of**traditions

**,**or

**,**as Barthes puts it

**,**"

**the**text

__is__

**a**tissue

**of**quotations drawn from

**the**innumerable centres

**of**culture "; it

__is__never original.

__Every__year

**,**

**on**

**the**last Sunday in April

**,**there

__is__

**an**ice fishing competition in

**the**frozen estuarine waters

**of**

**the**Anadyr River's mouth.

__Every__lattice element

**of**

**the**structure

__is__in

**its**proper place

**,**whether it

__is__

**a**single atom or

**a**molecular grouping.

0.187 seconds.