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Every octonion is a real linear combination of the unit octonions:

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## Some Related Sentences

Every and is

__Every__legislator from Brasstown Bald to Folkston

__is__going to have his every vote subjected to

**the**closest scrutiny as

**a**test

**of**his political allegiances, not his convictions.

__Every__detail in his interpretation has been beautifully thought out, and

**of**these I would especially cite

**the**delicious laendler touch

**the**pianist brings to

**the**fifth variation ( an obvious indication that he

__is__playing with Viennese musicians ), and

**the**gossamer shading throughout.

__Every__taxpayer

__is__well aware

**of**

**the**vast size

**of**our annual defense budget and most

**of**our readers also realize that

**a**large portion

**of**these expenditures go for military electronics.

__Every__few days, in

**the**early morning, as

**the**work progressed, twenty men would appear to push it ahead and to shift

**the**plank foundation that distributed its weight widely on

**the**Rotunda pavement, supported as it

__is__by ancient brick vaulting.

__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__library borrower, or at least those whose taste goes beyond

**the**five-cent fiction rentals, knows what it

__is__to hear

**the**librarian say apologetically, `` I'm sorry, but we don't have that book.

__Every__community, if it

__is__alive has

**a**spirit, and that spirit

__is__

**the**center

**of**its unity and identity.

The restricted principle "

__Every__partially ordered set has**a**maximal totally ordered subset "__is__also equivalent to AC over ZF.__Every__natural-born citizen

**of**

**a**foreign state who

__is__also an American citizen and every natural-born American citizen who

__is__

**a**citizen

**of**

**a**foreign land owes

**a**double allegiance, one to

**the**United States, and one to his homeland ( in

**the**event

**of**an immigrant becoming

**a**citizen

**of**

**the**US ), or to his adopted land ( in

**the**event

**of**an emigrant natural born citizen

**of**

**the**US becoming

**a**citizen

**of**another nation ).

__Every__line

**of**written text

__is__

**a**mere reflection

**of**references from any

**of**

**a**multitude

**of**traditions, or, as Barthes puts it, "

**the**text

__is__

**a**tissue

**of**quotations drawn from

**the**innumerable centres

**of**culture "; it

__is__never original.

__Every__year, on

**the**last Sunday in April, there

__is__an ice fishing competition in

**the**frozen estuarine waters

**of**

**the**Anadyr River's mouth.

__Every__lattice element

**of**

**the**structure

__is__in its proper place, whether it

__is__

**a**single atom or

**a**molecular grouping.

Every and real

*

__Every____real__Banach algebra which**is****a**division algebra**is**isomorphic to**the**reals,**the**complexes, or**the**quaternions.
*

__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 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__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__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.

__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.

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