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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.

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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__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__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 Hausdorff

*

__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__locally compact regular**space****is**completely regular**,****and**therefore every locally compact__Hausdorff__**space****is**Tychonoff**.**__Every__compact

__Hausdorff__

**space**

**is**also locally compact

**,**

**and**many examples of compact spaces may be found in the article compact

**space**

**.**

*

__Every__compact__Hausdorff__**space**of weight at most ( see Aleph number )**is**the continuous image of ( this does not need the continuum hypothesis**,**but**is**less interesting in its absence ).

Every and topological

*

__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__**finite**__topological__**space**gives rise to**a**preorder**on**its points**,**in which x ≤ y if**and**only if x belongs to every neighborhood of y**,****and**every**finite**preorder can be formed as the specialization preorder of**a**__topological__**space**in this way**.**__Every__group can be trivially made into

**a**

__topological__group

**by**considering it with the discrete

**topology**; such groups are called discrete groups

**.**

__Every__

__topological__group can be viewed as

**a**uniform

**space**in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps

**.**

__Every__subgroup of

**a**

__topological__group

**is**itself

**a**

__topological__group when given the subspace

**topology**

**.**

__Every__

__topological__ring

**is**

**a**

__topological__group ( with respect to addition )

**and**hence

**a**uniform

**space**in

**a**natural manner

**.**

*

__Every__non-empty Baire**space****is**of second category in itself**,****and**every intersection of countably many dense open subsets of X**is**non-empty**,**but the converse of neither of these**is**true**,**as**is**shown**by**the__topological__disjoint sum of the rationals**and**the unit interval 1**.**__Every__directed acyclic graph has

**a**

__topological__ordering

**,**an ordering of the vertices such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge

**.**

__Every__Boolean

**algebra**can be obtained in this way from

**a**suitable

__topological__

**space**: see Stone's representation theorem for Boolean algebras

**.**

__Every__such regular cover

**is**

**a**principal G-bundle

**,**where G = Aut ( p )

**is**considered as

**a**discrete

__topological__group

**.**

__Every__Boolean

**algebra**

**is**

**a**Heyting

**algebra**when

**a**→ b

**is**defined as usual as ¬

**a**∨ b

**,**as

**is**every complete distributive lattice when

**a**→ b

**is**taken to be the supremum of the set of all c for which

**a**∧ c ≤ b

**.**The open sets of

**a**

__topological__

**space**form

**a**complete distributive lattice

**and**hence

**a**Heyting

**algebra**

**.**

__Every__

__topological__group

**is**an H-space ; however

**,**in the general case

**,**as compared to

**a**

__topological__group

**,**H-spaces may lack associativity

**and**inverses

**.**

__Every__interior

**algebra**can be represented as

**a**

__topological__field of sets with its interior

**and**closure operators corresponding to those of the

__topological__

**space**

**.**

__Every__metric

**space**which

**is**ccc

**is**also separable

**,**but in general

**a**ccc

__topological__

**space**need not be separable

**.**

__Every__locally compact group which

**is**second-countable

**is**metrizable as

**a**

__topological__group ( i

**.**e

**.**can be given

**a**left-invariant metric compatible with the

**topology**)

**and**complete

**.**

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

**.**

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