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* 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.

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## Some Related Sentences

Every and unital

*****

__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 and real

*****

__Every__

__real__

**Banach**

**algebra**

**which**

**is**a division

**algebra**

**is**

**isomorphic**

**to**

**the**

**reals**

**,**

**the**

**complexes**

**,**

**or**

**the**

**quaternions**

**.**

__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

**.**

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