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Page "Lie group" ¶ 104
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Every and vector
** 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 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 and v
Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.
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 Vitali set V is uncountable, and v − u is irrational for any.
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Every vector space V with seminorm p ( v ) induces a normed space V / W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p ( v )
When a variable, v, is on the LHS of an assignment statement, such as s ( j ), then s ( j ) is a definition of v. Every variable ( v ) has at least one definition by its declaration ( V ) ( or initialization ).

Every and determines
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 action has a reaction and the force determines one's next incarnation.
Every number is thought of as a decimal fraction with the initial decimal point omitted, which determines the filing order.
: Every host in the network must have a unique address that determines where it is.
Every structure is associated with a certain quantity of energy, which determines the stability of the molecule or ion ( the lower energy, the greater stability ).
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 determines one probability distribution for.
Every two years, the MSHSL determines a school's activity classification and section placement.
Every character has three stats which help him in battle: Brawn stat determines weapon damage, Brains affects the effectiveness of super powers, and Toughness is the character's defense.
* Every pair of distinct points determines a unique line.
* Every non-collinear set of n points determines at least n distinct lines.
Every March, a 68-team, six-round, single-elimination tournament ( commonly called March Madness ) determines the national champions of NCAA Division I men's college basketball.

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 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 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 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 real m-by-n matrix yields a linear map from R < sup > n </ sup > to R < sup > m </ sup >.
* Every ( biregular ) algebraic automorphism of a projective space is projective linear.
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.
* Every irreducible polynomial in K which has a root in L factors into linear factors in L.
Every linear function on a finite-dimensional space is continuous.

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