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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.

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## Some Related Sentences

Every and normed

__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__bounded linear transformation from

**a**

__normed__

**vector**

**space**to

**a**complete,

__normed__

**vector**

**space**can be

**uniquely**extended to

**a**bounded linear transformation from

**the**

**completion**

**of**to

**.**

__Every__

**vector**

**space**

**V**with seminorm p ( v ) induces

**a**

__normed__

**space**

**V**/ W,

**called**

**the**quotient

**space**, where W

**is**

**the**

**subspace**

**of**

**V**consisting

**of**all vectors v in

**V**with p ( v )

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__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__

__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 V

__Every__pope from Miltiades occupied

**the**Lateran Palace until

**the**reign

**of**

**the**French Pope Clement

__V__, who in 1309 decided to transfer

**the**official seat

**of**

**the**Catholic Church to Avignon,

**a**papal fief that was an enclave within France

**.**

__Every__measurable cardinal κ

**is**

**a**0-huge cardinal because < sup > κ </ sup > M ⊂ M, that

**is**, every function from κ to M

**is**in M

**.**Consequently,

__V__< sub > κ + 1 </ sub >⊂ M

**.**

*

__Every__cover**is****a**local homeomorphism — that**is**, for every, there exists**a**neighborhood**of**c**and****a**neighborhood**of**such that**the**restriction**of**p to U yields**a**homeomorphism from U to__V__**.**This implies that C**and**X share all local properties**.**__Every__graph has

**a**harmonious coloring, since it suffices to assign every vertex

**a**distinct color

**;**thus χ < sub > H </ sub >( G ) ≤ |

__V__( G )|.

Indeed, in

**a**2011 review**of**" I Want My MTV "**by**Craig Marks**and**Rob Tannenbaum, Dwight Garner recalled: "__Every__sentient straight male in**the**country developed**a**schoolboy crush on Martha Quinn, one**of****the**first__V__**.**J**.**
Iris Adrian was an only child, raised

**by****a**single mother, Florence__V__**.**Hostetter ( née__Every__), in Los Angeles, California**.**__Every__year,

**the**Blue Cross Arena hosts regular season Section

__V__high school hockey games

**and**

**the**Section

__V__Basketball championship

**.**

When

**a**variable, v,**is**on**the**LHS**of**an assignment statement, such**as**s ( j ), then s ( j )**is****a**definition**of**v**.**__Every__variable ( v ) has at least one definition**by**its declaration (__V__) ( or initialization ).
King's Quest

__V__also inspired**a**text based remake, King's Quest V-The Text Aventure,**and**also another parody fan-game Owl's Quest:__Every__Owl has it's Day starring Cedric which pokes fun**of**many**of****the**situations**and**mannerisms**of**Cedric**.**__Every__Man Out

**of**His Humour includes several references to Shakespeare

**and**his contemporaneous works:

**a**mention

**of**Justice Silence from Henry IV, Part 2 —"

**this**

**is**

**a**kinsman to Justice Silence " (

__V__, ii )

**and**two allusions to Julius Casear, which help to date that play to 1599

**.**

" Et tu, Brute " occurs in

__V__, iv**of**__Every__Man Out**;**in III, i appears " reason long since**is**fled to animals ,"**a**paraphrase**of**Shakespeare's line " O judgment, thou art fled to brutish beasts " in Julius Caesar, III, ii, 104**.**0.917 seconds.