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.

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

The definition of permanent agriculture as that which can be sustained indefinitely was supported by Australian P. A. Yeomans in his 1973 book Water for Every Farm.

Every vertex in a polygon is assigned a texture coordinate ( which in the 2d case is also known as a UV coordinate ) either via explicit assignment or by procedural definition.

Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

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.

Note: Every cyclic permutation of definition type 1 will be constructed

Note: Every permutation over a set with k elements is a cyclic permutation of definition type 2 if and only if it is a cyclic permutation of definition type 1 with gcd ( k, offset )

Note: Every cyclic permutation of definition type 3 may be seen as an union of a cyclic permutation of definition type 2 and some fixed points.

Every cyclic permutation of definition type 2 may be seen ″ as a cyclic permutation of definition type 3 with zero fixed points.

* Every variable is assumed to have a definition in the context or scope.

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 subset of a complete Boolean algebra has a supremum, by definition ; it follows that every subset also has an infimum ( greatest lower bound ).

Every discipline that uses residence time in some way adapts the definition in order to make it more specific to the application to which it is referring.

Maxwell Fyfe brought up Raeder's order of 15 October 1939, which read: " Measures which are considered necessary from a military point of view will have to be carried out, even if they are not covered by existing international law ... Every protest from neutral powers will have to be turned down ... The more ruthlessly economic warfare is waged ... the sooner the war will come to an end ".

Every Victoria line station apart from Pimlico was built as an interchange station, and several existing stations were rearranged to allow for cross-platform interchange with the new line.

On November 7, 2006 he also produced Lumines II for PSP, the sequel to the popular original ( this time supplementing the original score with a heavy integration of music videos ), and a shooter / puzzle hybrid game called Every Extend Extra ( a heavily-modified ' synesthesia ' update to an existing PC freeware title, Every Extend ," by indie developer " Omega "), which was released on August 7, 2006.

Every five years the NZ Transport Agency will embark on a state highway review to consider whether the existing network should be expanded or reduced, according to traffic flows, changes in industry, tourism and development.

Every new bee strain or bee race was first crossed with the existing Buckfast Bee.

*" Every denunciation of existing law tends in some measure to increase the probability that there will be violation of it.

It is arguable that the recent Consultation Paper issued by CAFCASS and entitled Every Day Matters is a recognition of the hurt and resentment caused by the existing Family Law system, and is a call for radical change, particularly by the early adoption of Shared Residence in the vast majority of cases.

Every time a file is saved, rather than overwriting the existing version, a new file with the same name but an incremented version number is created.

Every seignory now existing must have been created before the statute Quia Emptores ( 1290 ), which forbade the future creation of estates in fee-simple by subinfeudation.

Every particular finite line, surface, or solid which may possibly be the object of our thought is an idea existing only in the mind, and consequently each part of it must be perceived.

Every year the existing systems are improving, new departments are going to be opened.

Every major European car manufacturer has a panel van in their line-up ; these models used to be modified versions of existing passenger cars, such as the Citroën Visa-based Citroën C15 or the SEAT Ibiza-based SEAT Inca.

Every community college evaluate each of its existing education programs and services on a five-year cycle to ensure that the programs continue to be justified on the basis of need, quality, and cost.

Every time a FairPlay-protected track is copied onto the iPod, iTunes will copy the user key from its own key repository to the key repository on the iPod.

Every record has a unique identifier, which indicates the domain name of the original repository where the record was created.

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.