** Every vector space has a basis.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

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 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 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 finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

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 vector in the space may be written as a linear combination of unit vectors.

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 coalgebra, by ( vector space ) duality, gives rise to an algebra, but not in general the other way.

* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.

For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted.

Every vector space is free, and the free vector space on a set is a special case of a free module on a set.

** Every field has an algebraic closure.

** Every field extension has a transcendence basis.

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 field is, of course, one-dimensional over its center.

Every time an MTA receives an email message, it adds a < tt > Received </ tt > trace header field to the top of the header of the message, thereby building a sequential record of MTAs handling the message.

Every subfield of an ordered field is also an ordered field in the inherited order.

Every ordered field contains an ordered subfield that is isomorphic to the rational numbers.

Every ordered field is a formally real field.

Every subfield of an ordered field is also an ordered field ( inheriting the induced ordering ).

Every ordered field can be embedded into the surreal numbers.

Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares.

Every instruction consists of a 9-bit opcode, a 4-bit register code, and a 23-bit effective address field, which consists in turn of a 1-bit indirect bit, a 4-bit register code, and an 18-bit offset.

Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations.

Every planetary body ( including the Earth ) is surrounded by its own gravitational field, which exerts an attractive force on all objects.

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every local field is isomorphic ( as a topological field ) to one of the following:

Every year, the British Political Studies Association awards the Walter Bagehot Prize for the best dissertation in the field of government and public administration.

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form

Every kind of material has unique magnetic properties, even those that we do not think of as being “ magnetic .” Different materials below the ground can cause local disturbances in the Earth ’ s magnetic field that are detectable with sensitive magnetometers.

Every Vitali set V is uncountable, and v − u is irrational for any.

Every function that was not assigned to a key could be invoked through the < u > XEQ </ u > key ( pronounced EXEQTE — " execute ") and spelled out in full, e. g. < u > XEQ </ u > FACT for the factorial function.

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 set S can be turned into a pointed dcpo by adding a least element ⊥ and introducing a flat order with ⊥ ≤ s and s ≤ s for every s ∈ S and no other order relations.

# Every subset of an independent set is independent, i. e., for each E ' ⊆ E, E ∈ I → E ' ∈ I.

Every car uses either a 5. 0 L Ford " Boss 302 " SVO or a 5. 0 L Chevrolet small block race-engine ( depending on the make )-capable of producing between 460 and 485 kW ( 620 — 650 bhp ) of power, but generally quoted as a little over 450 kW ( 600 bhp ) in race trim.

# " Every Little Bit o ' You " ( Irwin Levine, L. Russell Brown )

** L. A. Reid and Babyface for " Every Little Step " performed by Bobby Brown

Every year, usually around August, this marina hosts the R. N. L. I.

Every musician in L. A. wanted to participate, but soon Lennon's drinking and Spector's erratic behavior ( which included his firing a gun in the studio control room ) caused the sessions to break down.

* 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.

* Every irreducible polynomial in K which has a root in L factors into linear factors in L.

Every year CEI hosts an annual dinner gala and presents the Julian L. Simon Memorial Award.

** Every single member of the New Covenant " knows the L " in an intimate way.

Every non-trivial problem in L is complete under log-space reductions so weaker reductions are required to identify meaningful notions of L-completeness, the most common being first-order reductions.

Russell L. Ackoff wrote about complex problems as messes: " Every problem interacts with other problems and is therefore part of a set of interrelated problems, a system of problems ….

Professor Herman Bondi, who gave the first J. L. Synge Public Lecture in 1992, had this to say: " Every one of the other 88 % has been deeply influenced by his geometric vision and the clarity of his expression ".

Every L. A. # 1 song would be played in chronological order ( utilizing the older KHJ radio charts and other local charts ) from 1955 through 1985.

Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L < sup >∞</ sup >( X ) for some standard measure space ( X, μ ) and conversely, for every standard measure space X, L < sup >∞</ sup >( X ) is a von Neumann algebra.

Every Data Matrix is composed of two solid adjacent borders in an " L " shape ( called the " finder pattern ") and two other borders consisting of alternating dark and light " cells " or modules ( called the " timing pattern ").

Every year since 1994, Lloydies have climbed onto the top of Millikan Library to construct the Lloyd Christmas Tree, a monumental structure of numerous Christmas lights strung together to resemble a 10-story Christmas tree topped with a 12-feet-tall " L ." The latest off-campus pranks have been the pranking of MIT's Campus Preview Weekend, which included many Lloydies.

* James Allen ’ s Book of Meditations for Every Day in the Year ( 1913 ) of earlier texts, compiled by his wife Lily L. Allen