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* Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.

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## Some Related Sentences

Every and real

*****

__Every__unital

__real__

**Banach**

**algebra**with no zero divisors

**,**and in

**which**every principal ideal

**is**closed

**,**

**is**

**isomorphic**

**to**

**the**

**reals**

**,**

**the**

**complexes**

**,**

**or**

**the**

**quaternions**

**.**

*****

__Every__commutative

__real__unital Noetherian

**Banach**

**algebra**with no zero divisors

**is**

**isomorphic**

**to**

**the**

__real__

**or**complex numbers

**.**

*****

__Every__commutative

__real__unital Noetherian

**Banach**

**algebra**( possibly having zero divisors )

**is**finite-dimensional

**.**

__Every__sequence that ran off

**to**infinity in

**the**

__real__line will then converge

**to**∞ in this compactification

**.**

__Every__holomorphic function can be separated into its

__real__and imaginary parts

**,**and each of these

**is**

**a**solution of Laplace's equation on R < sup > 2 </ sup >.

__Every__ordered field

**is**

**a**formally

__real__field

**,**i

**.**e., 0 cannot be written as

**a**sum of nonzero squares

**.**

*****

__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__non-negative

__real__number

**a**has

**a**unique non-negative square root

**,**called

**the**principal square root

**,**

**which**

**is**denoted by

**,**where √

**is**called

**the**radical sign

**or**radix

**.**

__Every__nonzero

__real__number has

**a**multiplicative inverse ( i

**.**e

**.**an inverse with respect

**to**multiplication ) given by (

**or**).

__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__Riemann surface

**is**

**a**two-dimensional

__real__analytic manifold ( i

**.**e.,

**a**surface ), but it contains more structure ( specifically

**a**complex structure )

**which**

**is**needed for

**the**unambiguous definition of holomorphic functions

**.**

In his book Nirvana: The Stories Behind

__Every__Song**,**Chuck Crisafulli writes that**the**song " stands out in**the**Cobain canon as**a**song with**a**very specific genesis and**a**very__real__subject ".__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 and Banach

__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__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__separable metric space

**is**isometric

**to**

**a**subset of C (),

**the**separable

__Banach__space of continuous functions → R

**,**with

**the**supremum norm

**.**

Every and algebra

__Every__associative

__algebra__

**is**obviously alternative

**,**but so too are some strictly nonassociative algebras such as

**the**octonions

**.**

__Every__Boolean

__algebra__( A

**,**∧, ∨) gives rise

**to**

**a**ring ( A

**,**+, ·) by defining

**a**+ b := (

**a**∧ ¬ b ) ∨ ( b ∧ ¬

**a**) = (

**a**∨ b ) ∧ ¬(

**a**∧ b ) ( this operation

**is**called symmetric difference in

**the**case of sets and XOR in

**the**case of logic ) and

**a**· b :=

**a**∧ b

**.**The zero element of this ring coincides with

**the**0 of

**the**Boolean

__algebra__;

**the**multiplicative identity element of

**the**ring

**is**

**the**1 of

**the**Boolean

__algebra__

**.**

__Every__continuous map f: X → Y induces an

__algebra__homomorphism C ( f ): C ( Y ) → C ( X ) by

**the**rule C ( f )( φ ) = φ o f for every φ in C ( Y ).

__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__associative

__algebra__

**is**obviously power-associative

**,**but so are all other alternative algebras ( like

**the**octonions

**,**

**which**are non-associative ) and even some non-alternative algebras like

**the**sedenions

**.**

__Every__random vector gives rise

**to**

**a**probability measure on R < sup > n </ sup > with

**the**Borel

__algebra__as

**the**underlying sigma-algebra

**.**

__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__Boolean

__algebra__can be obtained in this way from

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

**.**

__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__Heyting

__algebra__with exactly one coatom

**is**subdirectly irreducible

**,**whence every Heyting

__algebra__can be made an SI by adjoining

**a**new top

**.**

Every and which

__Every__new scandal

__which__would provide more `` copy '' for Marshall's pen would thus mean more publicity for Welch

**.**

__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__family of Riviera Presbyterian Church has been asked

**to**read

**the**Bible and pray together daily during National Christian Family Week and

**to**undertake one project in

__which__all members of

**the**family participate

**.**

** 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**.*******

__Every__continuous functor on

**a**small-complete category

__which__satisfies

**the**appropriate solution set condition has

**a**left-adjoint (

**the**Freyd adjoint functor theorem ).

__Every__telephone company

**,**whether large

**or**small

**,**determines its own ANAC for each individual central office

**,**

__which__tends

**to**perpetuate

**the**current situation of

**a**mess of overlapping and /

**or**spotty areas of coverage

**.**

__Every__field has an algebraic extension

__which__

**is**algebraically closed ( called its algebraic closure ), but proving this in general requires some form of

**the**axiom of choice

**.**

__Every__year since 1972

**the**BVI has hosted

**the**Spring Regatta

**,**

__which__

**is**

**a**seven-day collection of sailing races throughout

**the**islands

**.**

__Every__character

**is**automatically continuous from A

**to**C

**,**since

**the**kernel of

**a**character

**is**

**a**maximal ideal

**,**

__which__

**is**closed

**.**

__Every__computer contains an internal clock that regulates

**the**rate at

__which__instructions are executed and synchronizes all

**the**various computer components

**.**

*****Duality:

__Every__statement

**,**theorem

**,**

**or**definition in category theory has

**a**dual

__which__

**is**essentially obtained by " reversing all

**the**arrows ".

__Every__context-sensitive grammar

__which__does not generate

**the**empty string can be transformed into an equivalent one in Kuroda normal form

**.**

According

**to**__Every__**,**one example may be "**the**myth of St**.**George " and other stories about saints battling dragons**,**__which__were " modelled no doubt in many cases on older representations of**the**creator and preserver of**the**world in combat with chaos ".__Every__argues that "

**the**disparagement of myth in our own civilization " stems partly from objections

**to**perceived idolatry

**,**objections

__which__intensified in

**the**Reformation

**,**both among Protestants and among Catholics reacting against

**the**classical mythology revived during

**the**Renaissance

**.**

__Every__grammar in Chomsky normal form

**is**context-free

**,**and conversely

**,**every context-free grammar can be transformed into an equivalent one

__which__

**is**in Chomsky normal form

**.**

__Every__four years

**,**during

__which__an extra 24 hours have accumulated

**,**one extra day

**is**added

**to**keep

**the**count coordinated with

**the**sun's apparent position

**.**

__Every__sequence can

**,**thus

**,**be read in three reading frames

**,**each of

__which__will produce

**a**different amino acid sequence ( in

**the**given example

**,**Gly-Lys-Pro

**,**Gly-Asn

**,**

**or**Glu-Thr

**,**respectively ).

__Every__time

**the**boy had an injury

__which__caused him internal

**or**external bleeding

**,**

**the**Tsarina called on Rasputin

**,**and

**the**Tsarevich subsequently got better

**.**

__Every__week before June 19

**,**

**the**strip focuses on Garfield's birthday

**,**

__which__he dreads because of his fear of getting older

**.**

:::

__Every__Bill__which__shall have passed**the**House of Representatives and**the**Senate**,**shall**,**before it become**a**Law**,**be presented**to****the**President of**the**United States ; If he approves he shall sign it**,**but if not he shall return it**,**with his Objections**to**that House in__which__it shall have originated ...0.207 seconds.