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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.

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## Some Related Sentences

Every and subset

__Every__such

__subset__has

**a**

**smallest**element, so to specify our choice function we can simply say that it maps each

**set**to

**the**least element

**of**that

**set**

**.**

** Zorn's lemma:

__Every__non-empty partially ordered**set**in which every chain**(**i**.**e**.**totally ordered__subset__) has an upper bound contains at least one maximal element**.**
The restricted principle "

__Every__partially ordered**set**has**a**maximal totally ordered__subset__"**is**also equivalent to AC over ZF**.**
*

__Every__separable metric**space****is**isometric to**a**__subset__**of****the****(**non-separable ) Banach**space**l < sup >∞</ sup >**of****all**bounded real sequences with**the**supremum norm ; this**is**known as**the**Fréchet embedding**.**
*

__Every__separable metric**space****is**isometric to**a**__subset__**of**C (),**the**separable Banach**space****of**continuous functions → R, with**the**supremum norm**.**__Every__element s, except

**a**possible greatest element, has

**a**unique successor

**(**next element

**),**

**namely**

**the**least element

**of**

**the**

__subset__

**of**

**all**elements greater than s

**.**

__Every__

__subset__which has an upper bound has

**a**least upper bound

**.**

__Every__

__subset__

**of**

**a**nowhere dense

**set**

**is**nowhere dense, and

**the**union

**of**finitely many nowhere dense

**sets**

**is**nowhere dense

**.**

__Every__

__subset__

**of**

**the**Hilbert cube inherits from

**the**Hilbert cube

**the**properties

**of**being both metrizable

**(**and therefore T4 ) and second countable

**.**

It

**is**more interesting that**the**converse also holds:__Every__second countable T4**space****is**homeomorphic to**a**__subset__**of****the**Hilbert cube**.**
*

__Every__totally disconnected compact metric**space****is**homeomorphic to**a**__subset__**of****a**countable product**of**discrete spaces**.**
*

__Every__irreducible closed__subset__**of**P < sup > n </ sup >( k )**of**codimension one**is****a**hypersurface ; i**.**e.,**the**zero**set****of**some homogeneous polynomial**.**

Every and vector

__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__

__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 )0.326 seconds.