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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.

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## Some Related Sentences

Every and bounded

*

__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__finite or

__bounded__interval

**of**

**the**real numbers that contains an infinite number

**of**points must have at least one point

**of**accumulation

**.**

*

__Every__totally ordered set that is**a**__bounded__lattice is also**a**Heyting algebra**,**where is equal**to**when**,**and 1 otherwise**.**
This is really

**a**special case**of****a**more general fact:__Every__continuous function**from****a**compact**space**into**a**metric**space**is__bounded__**.**
For example

**,****to**study**the**theorem “__Every____bounded__sequence**of**real numbers has**a**supremum ” it is necessary**to**use**a**base system which**can**speak**of**real numbers and sequences**of**real numbers**.**__Every__maximal outerplanar graph with n vertices has exactly 2n − 3 edges

**,**and every

__bounded__face

**of**

**a**maximal outerplanar graph is

**a**triangle

**.**

__Every__three years

**the**Company makes an award

**to**

**the**three buildings or structures

**,**in

**the**area

__bounded__by

**the**M25 motorway

**,**which respectively embody

**the**most outstanding example

**of**brickwork

**,**

**of**slated or tiled roof and

**of**hard-surface tiled wall and / or floor

**.**

__Every__Polish town was

__bounded__

**to**put up

**a**quantity

**of**soldiers-this was

**a**conspicuous sign

**of**

**a**power

**of**

**a**given town how much soldiers it had

**to**put up

**.**

__Every__

**complete**lattice is also

**a**

__bounded__lattice

**,**which is

**to**say that it has

**a**greatest and least element

**.**

__Every__

__bounded__positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0

**.**improved this criterion by showing that it is sufficient

**to**ask that

**,**for every continuous positive-definite compactly supported function f on G

**,**

**the**function Δ < sup >– ½ </ sup > f has non-negative integral with respect

**to**Haar measure

**,**where Δ denotes

**the**modular function

**.**

__Every__subset

**of**

**a**totally

__bounded__

**space**is

**a**totally

__bounded__set ; but even if

**a**

**space**is not totally

__bounded__

**,**some

**of**its subsets still will

**be**

**.**

__Every__topological

**vector**

**space**X gives

**a**bornology on X by defining

**a**subset

**to**

**be**

__bounded__iff for all open sets containing zero there exists

**a**with

**.**

__Every__(

__bounded__) convex polytope is

**the**image

**of**

**a**simplex

**,**as every point is

**a**convex combination

**of**

**the**( finitely many ) vertices

**.**

Every and linear

__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__time

**a**diode switches

**from**on

**to**off or vice versa

**,**

**the**configuration

**of**

**the**

__linear__network changes

**.**

__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__smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural

__linear__maps between

**the**corresponding tangent spaces:

*

__Every____linear__combination**of**its components Y =**a**< sub > 1 </ sub > X < sub > 1 </ sub > + … +**a**< sub > k </ sub > X < sub > k </ sub > is normally distributed**.**__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__sedenion is

**a**real

__linear__combination

**of**

**the**unit sedenions 1

**,**< var > e </ var >< sub > 1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >,

__Every__physical quantity has

**a**Hermitian

__linear__operator associated

**to**it

**,**and

**the**states where

**the**value

**of**this physical quantity is definite are

**the**eigenstates

**of**this

__linear__operator

**.**

__Every__nontrivial proper rotation in 3 dimensions fixes

**a**unique 1-dimensional

__linear__subspace

**of**R < sup > 3 </ sup > which is called

**the**axis

**of**rotation ( this is Euler's rotation theorem ).

__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 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__

**vector**

**a**in three dimensions is

**a**

__linear__combination

**of**

**the**standard basis vectors i

**,**j

**,**and k

**.**

__Every__lattice in

**can**

**be**generated

**from**

**a**basis for

**the**

**vector**

**space**by forming all

__linear__combinations with integer coefficients

**.**

__Every__

__linear__program has

**a**dual problem with

**the**same optimal solution

**,**but

**the**variables in

**the**dual problem correspond

**to**constraints in

**the**primal problem and vice versa

**.**

Every and transformation

__Every__element in O ( 1

**,**3 )

**can**

**be**written as

**the**semidirect product

**of**

**a**proper

**,**orthochronous

__transformation__and an element

**of**

**the**discrete group

__Every__universal cover p: D → X is regular

**,**with deck

__transformation__group being isomorphic

**to**

**the**fundamental group

**.**

__Every__six rounds

**,**

**a**logical

__transformation__layer is applied:

**the**so-called " FL-function " or its inverse

**.**

__Every__orthogonal

__transformation__

**of**

**a**k-frame in R < sup > n </ sup > results in another k-frame

**,**and any two k-frames are related by some orthogonal

__transformation__

**.**

__Every__element x

**of**G gives rise

**to**

**a**tensor-preserving self-conjugate natural

__transformation__via multiplication by x on each representation

**,**and hence one has

**a**map

**.**

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