* 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 real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as

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.

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 dual number has the form z = a + bε with a and b uniquely determined real numbers.

Every real number has an additive inverse ( i. e. an inverse with respect to addition ) given by.

Every nonzero real number has a multiplicative inverse ( i. e. an inverse with respect to multiplication ) given by ( or ).

Every real number, rational or not, is equated to one and only one cut of rationals.

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 octonion is a real linear combination of the unit octonions:

Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.

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 real number greater than zero or every complex number except 0 has two square roots.

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 node has a location, which is a number between 0 and 1.

The album's lead single, " She's Every Woman " peaked at number-one on the Billboard Country Chart, however its follow-up single, " The Fever " ( a cover of an Aerosmith song ) only peaked at number 23, becoming Brooks's first released Country single to not chart on the Top 10.

Every rational number has a unique representation as an irreducible fraction.

Every year the International Labour Conference's Committee on the Application of Standards examines a number of alleged breaches of international labour standards.

Every year from 1985 through 1993, the number of attendees tripled.

Every LORAN chain in the world uses a unique Group Repetition Interval, the number of which, when multiplied by ten, gives how many microseconds pass between pulses from a given station in the chain.

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 ketose will have 2 < sup >( n-3 )</ sup > stereoisomers where n > 2 is the number of carbons.

Every aldose will have 2 < sup >( n-2 )</ sup > stereoisomers where n > 2 is the number of carbons.

The second single from the UK release was " With Every Heartbeat ", released in late July and reached number one on the UK singles chart.

# Every simple path from a given node to any of its descendant leaves contains the same number of black nodes.

Every positive number a has two square roots:, which is positive, and, which is negative.

Every number is thought of as a decimal fraction with the initial decimal point omitted, which determines the filing order.

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set.

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3.

# Every noun belongs to a unique number class.

* Hardy and Littlewood listed as their Conjecture I: " Every large odd number ( n > 5 ) is the sum of a prime and the double of a prime.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

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 corporation, whether financial or union, as well as every division of the administration, were set up as branches of the party, the CEOs, Union leaders, and division directors being sworn-in as section presidents of the party.

Every view of reality that is introduced in the story is later derided in some way, whether that view is traditional or iconoclastic.

Concerning the role of clave in salsa music, Charley Gerard states: “ The clave feeling is in the music whether or not the claves are actually being played .” Every ostinato part which spans a cycle of four main beats, has a specific alignment with clave, and expresses the rhythmic qualities of clave either explicitly or implicitly.

Every two years, a panel will determine whether a property in the rural district shall be absorbed into the urban district.

Every woman, being with child, who, with intent to procure her own miscarriage, shall unlawfully administer to herself any poison or other noxious thing, or shall unlawfully use any instrument or other means whatsoever with the like intent, and whosoever, with intent to procure the miscarriage of any woman whether she be or be not with child, shall unlawfully administer to her or cause to be taken by her any poison or other noxious thing, or unlawfully use any instrument or other means whatsoever with the like intent, shall be guilty of felony, and being convicted thereof shall be liable.

Every proclamation ( such as the announcement of each country during the parade of nations in the opening ceremony ) is spoken in these three languages, or the main two depending on whether the host country is an English or Spanish speaking country.

Every five years the NZ Transport Agency will embark on a state highway review to consider whether the existing network should be expanded or reduced, according to traffic flows, changes in industry, tourism and development.

Every smallest movement is placed with fierce pride ; at any moment it may break into such a frenzied gallop that you hardly know whether to feel exhilarated or terrified ".

Every day for instance, people are faced with the dilemma of whether or not to lie in a given situation.

Every person who knew him, whether of his own community or not, held him in the highest esteem.

Every shell, whether it be a high-explosive or shrapnel shell, was fixed to a brass case which was automatically ejected when the breech was opened.

Every telephone company, whether large or small, determines its own ringback numbers for each individual central office, which tends to perpetuate the current situation of a mess of overlapping and / or spotty areas of coverage.

Every person ( whether a Fiji citizen or not ) is granted the right to communicate with any organ of government, at both the national and local level, in any of the three official languages, " either directly or through a competent interpreter.

Every film today, whether it be live-action, computer generated, or traditional hand-drawn animation is made up of hundreds of individual shots that are all placed together during editing to form the single film that is viewed by the audience.

Every male Athenian citizen, irrespective to birth, occupation, and with a few exceptions, economic status, had the right to wield power as an official or Council member and actively participate in the decision-making process at the Assembly whether or not he currently held any official position.

" Every winter evening between 1924 and 1956, KFI would deliver a frost report at 8 pm that would tell citrus farmers whether to turn on wind machines or light " smudge pots " to keep their orange and lemon groves from freezing.

Every act performed by someone in the usual way of things has some kind of reward attached whether it is financial, power, love, status or just feeling good about oneself.

Every network member is an active contributor to IRGC ’ s work, whether via the provision of funding, expertise and research work, or a combination of these.

Every U. S. President since George Washington has used covert action as a part of their broader foreign policy, whether Republican or Democrat, liberal or conservative.

Every door, whether it leads to nothing or a level boss, gives the player a question mark ('?

( 1 ) Every one who, with intent to procure the miscarriage of a female person, whether or not she is pregnant, uses any means for the purpose of carrying out his intention is guilty of an indictable offence and liable to imprisonment for life.

Every doctor and dentist should, where patients are suffering from unclear pathological states and autoimmune diseases, consider whether side-effects from mercury released from amalgam may be one contributory cause of the symptoms.