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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.

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## 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__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__

__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__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 normed

__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__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 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__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__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__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 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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