* 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, whether integer, rational, or irrational, has a unique location on the line.

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 polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

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 pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

Every polynomial P in x corresponds to a function, ƒ ( x )

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

Every Boolean ring R satisfies x ⊕ x

Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )).

Every positive real number x has a single positive nth root, which is written.

Every non-zero number x, real or complex, has n different complex number nth roots including any positive or negative roots, see complex roots below.

Every such line meets the sphere of radius one centered in the origin exactly twice, say in P = ( x, y, z ) and its antipodal point (− x, − y, − z ).

Every locally constant function from the real numbers R to R is constant by the connectedness of R. But the function f from the rationals Q to R, defined by f ( x ) = 0 for x < π, and f ( x ) = 1 for x > π, is locally constant ( here we use the fact that π is irrational and that therefore the two sets

Every empty function is constant, vacuously, since there are no x and y in A for which f ( x ) and f ( y ) are different when A is the empty set.

Every real number x is surrounded by an infinitesimal " cloud " of hyperreal numbers infinitely close to it.

* 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 finitely generated ideal of A is principal ( i. e., A is a Bézout domain ) and A satisfies the ascending chain condition on principal ideals.

Every odd number q satisfies for.

Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is node pancyclic, meaning that for every vertex v and every k in the range from three to the number of vertices in the graph, there is a length-k cycle containing v. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not v, until the outer face of the remaining graph has length k.

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.

* Let K < sup > a </ sup > be an algebraic closure of K containing L. Every embedding σ of L in K < sup > a </ sup > which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.

Lemma ( Singleton bound ): Every linear code C satisfies.

# Every elementary substructure of a model of a theory T also satisfies T ; hence it is a submodel.