* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.

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 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 Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.

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 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 finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.

* Every finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C.

* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.

* Every finite-dimensional reflexive algebra is hyper-reflexive.

Every linear function on a finite-dimensional space is continuous.

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 form is the delimitation of a surface by another one ; it possesses an inner content, the effect it produces on one who looks at it attentively.

* Every open set E is inner regular:

Every inner automorphism is indeed an automorphism of the group G, i. e. it is a bijective map from G to G and it is a homomorphism ; meaning ( xy )< sup > a </ sup >

Every quadratic space is also an inner product space ( see below ).

Every smooth submanifold of R < sup > n </ sup > has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on R < sup > n </ sup >.

Every individual transmembrane segment of a protein also has topology, depending on location of its N-terminus on the inner or outer side of the membrane.

* Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ).

Every organism has an outer boundary which defines an Innenwelt or inner world and an Umwelt, surround-world or environment.

* Every Party member has a telescreen in his or her home, which the Thought Police uses to observe their actions and take note of anything that resembles an unorthodox opinion or an inner struggle.

Every Tacoma was manufactured with a composite inner bed that includes a deck rail system with four tie down cleats, hook-pins, storage boxes, and a power outlet ( models with TRD packages ).

According to Dale E. Dawkins, AMC's vice-president, " Every square inch of inner surface on exterior body panels is galvanized on our Spirit, Concord, and Eagle models.

Every PE keeps track of assigned inner label, and associates these with the VPLS instance.

Moreover, under the pseudonym ‘ Lt .- Col. William “ Bill ” Tanner ’ — M .’ s CoS and 007 ’ s best friend in SIS — Amis wrote his second Bond book, The Book of Bond, or Every Man His Own 007 ( 1965 ), a tongue-in-cheek, how-to-manual to help the everyman find his own inner secret agent.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

* 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 product is perfectly homogeneous and a perfect substitute for any other.

Every man-made product is composed of natural resources ( at its fundamental level ).

* Every product of ( arbitrarily many ) profinite groups is profinite ; the topology arising from the profiniteness agrees with the product topology.

Every software product has a target audience.

Every data source has a different data-access language ( or API ), driving up the costs to learn and use each vendor's product.

Every scene, every shot is the product of a keen and sensitive eye — an eye which has deep comprehension of the beauty of rugged people and a rugged world ".

Every composite number can be written as the product of two or more ( not necessarily distinct ) primes ; furthermore, this representation is unique up to the order of the factors.

Every third weekend in September Silver Creek hosts a Festival of Grapes to honor Concord grapes, an important agricultural product.

One drug company product manager has been quoted as saying, " Every marketer's dream is to find an unidentified or unknown market and develop it.

Every element in O ( 1, 3 ) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group

: Every product is subject to a life-cycle including a growth phase followed by a maturity phase and finally an eventual period of decline as sales falls.

* Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.

Every reactant in a spontaneous process has a tendency to form the corresponding product.

It is usually presented as sales minus sales discounts, returns, and allowances. Every time a business sells a product or performs a service, it obtains revenue.

Every reduced word is an alternating product of elements of G and elements of H, e. g.

* Every commutative semisimple ring must be a finite direct product of fields.