Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.

Any change to the information processing environment introduces an element of risk.

# Any pure element ( even if it forms diatomic molecules like chlorine, Cl < sub > 2 </ sub >) has an oxidation state ( OS ) of zero.

Any translating element was omitted, since it does not affect the FWHM.

Any document element could be given new " methods " ( capabilities ), and could respond to changes in the content or structure of the document itself.

Any monoid ( any algebraic structure with a single associative binary operation and an identity element ) forms a small category with a single object x.

* Any principal ideal domain, such as the integers, is Noetherian since every ideal is generated by a single element.

Any formal series with has a composition inverse provided is an invertible element of R. The coefficients are found recursively from the above formula for the coefficients of a composition, equating them with those of the composition identity X ( that is 1 at degree 1 and 0 at every degree greater than 1 ).

* Frozen: Any waterfall which has some element of ice.

Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices.

# Any group G may be regarded as an " abstract " category with one object,, and one morphism for each element of the group.

Any element from the holiday special that is not referenced in other works is considered S-Canon, which means that it is canon, and that it " happened ," but its canonicity is not set in stone.

Any set of sequences can be represented as paths in a tree, by forming a tree node for every prefix of a sequence and making the parent of one of these nodes represent the sequence with one fewer element ; the tree formed in this way for a set of strings is called a trie.

:: Theorem ( Lefschetz theorem on ( 1, 1 )- classes ) Any element of H < sup > 2 </ sup >( X, Z ) ∩ H < sup > 1, 1 </ sup >( X ) is the cohomology class of a divisor on X.

Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into k chains then the width of the order must be at most k. Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width.

Any point x gives rise to a continuous function p < sub > x </ sub > from the one element topological space 1 ( all subsets of which are open ) to the space X by defining p < sub > x </ sub >( 1 ) = x. Conversely, any function from 1 to X clearly determines one point: the element that it " points " to.

Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid.

* Any nonempty set S of submodules of the module has a maximal element ( with respect to set inclusion.

Any difference is amplified and used to control the regulation element in such a way as to reduce the voltage error.

Any element y ∈ Y := Ran ( S − I ) has a distance larger than or equal to 1 from u ( otherwise y < sub > i </ sub > = x < sub > i + 1 </ sub >-x < sub > i </ sub > would be positive and bounded away from zero, whence x < sub > i </ sub > could not be bounded ).

Any C *- algebra generated by a unitary element is the homomorphic image of this universal C *- algebra.

* Any directed set A may be made into a Cauchy space by declaring a filter F to be Cauchy if, given any element n of A, there is an element U of F such that U is either a singleton or a subset of the tail

He merely said, `` Any good decorator these days can make you a tasteful home ''.

Any alteration of one of these factors is distortion, although we generally use that word only for effects so pronounced that they can be stated quantitatively on the basis of standard tests.

Any reputable French interne can supply you with a dozen similar instances, and I'll presently recount a case out of my own personal experience, but, for the moment, let's resume our catalogue.

Any needy family living in San Francisco can obtain toys by writing to Christmas Toys, 676 Howard street, San Francisco 5, and listing the parent's name and address and the age and sex of each child in the family between the ages of 1 and 12.

Any breach of these laws is a criminal offence and can lead to fine of up to £ 5, 000 or six months in prison.

Any use of antibiotics can increase selective pressure in a population of bacteria to allow the resistant bacteria to thrive and the susceptible bacteria to die off.

* Any ring A can be considered as a Z-algebra in a unique way.

Any of these body styles can optionally incorporate a " cutaway ".

Any Latin-American country can join the 1980 Montevideo Treaty.

Any given bookmaker will weight their odds so that no one customer can cover all outcomes at a profit against their books.

Any node in the data structure can be reached by starting at root node and repeatedly following references to either the left or right child.

Any of these ways can be criticized for ...

* Any locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.

Any Sangiovese-based wine made according to the Chianti guidelines from these vineyards can be labelled and marked under the basic Chianti DOCG should the producer wish to use the designation.

Any units of angular measure can be used for declination, but it is customarily measured in the degrees ( ° ), minutes ( ' ), and seconds ( " ) of sexagesimal measure, with 90 ° equivalent to circle.

Any convex polyhedron can be distorted into a canonical form, in which a midsphere ( or intersphere ) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.

Any person who uses the Spanish title of " Doctor " ( or " Dr .") without being included in this Government database can be prosecuted for fraud.

Any player on the kicking team behind the kicker, and including the kicker, can recover the kick.

Any sound that can be sampled can be used.

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters.

# Any real number can be added to both sides.

# Any real number can be subtracted from both sides.

# Any real number can be multiplied to both sides.