** 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 Lie group is parallelizable, and hence an orientable manifold ( there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity )

* Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.

* Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.

* Every holomorphic line bundle on a projective variety is a line bundle of a divisor.

Every vector bundle admits a connection.

Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve.

* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".

Every well-constructed theory of education has at its center a philosophical anthropology.

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

While also refining Galen's erroneous theory of the pulse, Avicenna provided the first correct explanation of pulsation: " Every beat of the pulse comprises two movements and two pauses.

Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories.

Every one of the features encompassed by the theory still requires a reason for it to be maintained after hominids left the aquatic environment.

" Some also hold to the theory that Kidd conspired with Henry Every and Oak Island was used as a pseudo community bank between the two.

# Every " good " scientific theory is a prohibition: it forbids certain things to happen.

# Every genuine test of a theory is an attempt to falsify it, or to refute it.

: Every countable theory which is satisfiable in a model M, is satisfiable in a countable substructure of M.

: Every countable theory which is satisfiable in a model is also satisfiable in a countable model.

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

Every computer must be compatible with the principles of information theory, statistical thermodynamics, and quantum mechanics.

Is fundamental for the many-body theory that Every operator can be expressed in terms of annihilation and creation operators.

Every Skolem theory is model complete, i. e. every substructure of a model is an elementary substructure.

Every issue of the magazine opens with a description of The Skeptics Society and its mission statement, which is to explore subjects such as creationism, pyramid power, Bigfoot, pseudohistorical claims ( as in the examples of Holocaust denial and extreme Afrocentrism ), the use or misuse of theory and statistics, conspiracy theories, urban myths, witch-hunts, mass hysterias, genius and intelligence, and cultural influences on science, as well as controversies involving protosciences at the leading edge of established science, and even fads like cryonics and low-carb diets.

Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied.

Every quantum field theory satisfying certain technical assumptions about its S-matrix that has non-trivial interactions can only have a symmetry Lie algebra which is always a direct product of the Poincaré group and an internal group if there is a mass gap: no mixing between these two is possible.

Every major Korean neo-Confucianist shared Toegye's preoccupation with single-mindedness, which signalled new stress on praxis in the development of Korean neo-Confucianism: the fusion of the metaphysical and the physical is better brought about through action than speculation, important as theory might be.

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

Every theory with quantifier elimination is model complete.

Every theory which refuses to ascribe to God an attribute which is essential to a worthy conception of His character is anti-theistic.