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

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

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 finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram – Schmidt process.

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 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.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every taxpayer is well aware of the vast size of our annual defense budget and most of our readers also realize that a large portion of these expenditures go for military electronics.

Every single problem touched on thus far is related to good marketing planning.

Every few days, in the early morning, as the work progressed, twenty men would appear to push it ahead and to shift the plank foundation that distributed its weight widely on the Rotunda pavement, supported as it is by ancient brick vaulting.

Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available.

Every man in every one of these houses is a Night Rider.

Every library borrower, or at least those whose taste goes beyond the five-cent fiction rentals, knows what it is to hear the librarian say apologetically, `` I'm sorry, but we don't have that book.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Every infinite game in which is a Borel subset of Baire space is determined.

Every natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

Every line of written text is a mere reflection of references from any of a multitude of traditions, or, as Barthes puts it, " the text is a tissue of quotations drawn from the innumerable centres of culture "; it is never original.

Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every year, on the last Sunday in April, there is an ice fishing competition in the frozen estuarine waters of the Anadyr River's mouth.

Every lattice element of the structure is in its proper place, whether it is a single atom or a molecular grouping.

Every play is subject to booth review with coaches only having one challenge.

Every city or town that refused surrender and resisted the Mongols was subject to destruction.

Every vessel in coastal and offshore waters is subject to the International Regulations for Preventing Collisions at Sea ( the COLREGS ).

Every year YCA presents Louder Than A Bomb, the world's largest team-based youth slam and subject of a forthcoming documentary by the same name.

Every subject had an obligation to protect the king's peace and assist in the suppression of riots.

Every member of the community was subject to criticism by committee or the community as a whole, during a general meeting.

Every member of the bands or tribes receiving a land allotment is subject to laws of the state or territory in which they reside.

Every new Lodge must be warranted or chartered by a Grand Lodge, but is subject to its direction only in enforcing the published Constitution of the jurisdiction.

Every aspect of the Mishnaic text is treated as a subject of close investigation.

Every pre-abelian category is of course an additive category, and many basic properties of these categories are described under that subject.

§ 1: " Every subject expects from the territorial prince security and protection, so it is the duty of the territorial prince, the rights of subjects to determine clearly and to guide the way of the actions how it is needed by universal and special prosperity.

“ Every machine is constructed for the purpose of performing certain mechanical operations, each of which supposes the existence of two other things besides the machine in question, namely, a moving power, and an object subject to the operation, which may be termed the work to be done.

Every new boy who enters the school is given a two week period of time called " grace " when he is not fully subject to all school rules and is shown the ropes by an assigned boy in the year above called a " Shepherd ".

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 law or resolution having the force of law, shall relate to but one subject, and that shall be expressed in the title.

Every governor since Ronald Reagan in 1968 has been subject to a recall effort, but Gray Davis was the first governor whose opponents gathered the necessary signatures to qualify for a special election.

Every feature on every manufactured part is subject to variation, therefore, the limits of allowable variation must be specified.

Durbin signed a contract with MGM in 1935 and made her first film appearance in a short subject, Every Sunday ( 1936 ), with another young contract player, Judy Garland.

Every Java 2D operation is subject to a transform, so that shapes may be translated, rotated, sheared, and scaled as they are drawn.

In a 2004 interview, Kadivar told a journalist, " Every member of society and every member of government is subject to the law.

A universal affirmative proposition, such as, ' Every man is white ' and a universal negative proposition having the same subject and predicate, such as, ' No man is white ,' are called contrary.

Every subject or course successfully completed earns ‘ credits ’ towards the SACE.

He was the subject of the film Every Child is Born a Poet: The Life and Work of Piri Thomas, by Jonathan Robinson, which featured a soundtrack by Kip Hanrahan.

He may have been the subject of a number of Elizabethan satires such as Thomas Nashe's Lenten Stuffe, Ben Jonson's Every Man in his Humour, and may have been the model of Shakespeare's Falstaff, who was originally given the name ' Oldcastle '.