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* 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.

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## Some Related Sentences

Every and separable

*****

__Every__

__separable__

**metric**

**space**

**is**

**isometric**

**to**

**a**

**subset**

**of**C (),

**the**

__separable__

**Banach**

**space**

**of**continuous functions → R,

**with**

**the**

**supremum**

**norm**

**.**

*****

__Every__reduced commutative k-algebra A

**is**

**a**

__separable__algebra

**;**i

**.**e.,

**is**reduced for every field extension F / k

**.**

__Every__commutative von Neumann algebra on

**a**

__separable__Hilbert

**space**

**is**isomorphic

**to**L

**<**

**sup**

**>∞</**

**sup**>( X

**)**for some standard measure

**space**

**(**X, μ

**)**and conversely, for every standard measure

**space**X, L

**<**

**sup**

**>∞</**

**sup**>( X

**)**

**is**

**a**von Neumann algebra

**.**

__Every__

**metric**

**space**which

**is**ccc

**is**also

__separable__, but in general

**a**ccc topological

**space**need not be

__separable__

**.**

Every and metric

__Every__smooth submanifold

**of**R

**<**

**sup**

**>**n </

**sup**

**>**has an induced Riemannian

__metric__g:

**the**inner product on each tangent

**space**

**is**

**the**restriction

**of**

**the**inner product on R

**<**

**sup**

**>**n </

**sup**>.

__Every__building has

**a**canonical length

__metric__inherited from

**the**geometric realisation obtained by identifying

**the**vertices

**with**an orthonormal basis

**of**

**a**Hilbert

**space**

**.**

*****

__Every__totally disconnected compact

__metric__

**space**

**is**homeomorphic

**to**

**a**

**subset**

**of**

**a**countable product

**of**discrete spaces

**.**

This

**is**really**a**special case**of****a**more general fact:__Every__continuous function from**a**compact**space**into**a**__metric__**space****is****bounded****.**
#

__Every__Riemannian__metric__on**a**Riemann surface**is**Kähler, since**the**condition for ω**to**be closed**is**trivial in 2**(****real****)**dimensions**.**__Every__hyperkähler manifold M has

**a**2-sphere

**of**complex structures

**(**i

**.**e

**.**integrable almost complex structures

**)**

**with**respect

**to**which

**the**

__metric__

**is**Kähler

**.**

__Every__special uniformly continuous real-valued function defined on

**the**

__metric__

**space**

**is**uniformly approximable by means

**of**Lipschitz functions

**.**

__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 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__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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