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Every vector space is free, and the free vector space on a set is a special case of a free module on a set.

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## Some Related Sentences

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__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 ).

__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__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__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 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

**.**

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