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 collection of closed subsets of X with the finite intersection property has nonempty intersection.

* Any topologically closed subgroup of a Lie group is a Lie group.

* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.

Any force that passes the closed path test for all possible closed paths is classified as a conservative force.

Any intersection of closed sets is closed ( including intersections of infinitely many closed sets ), and any union of finitely many closed sets is closed.

Eric Raymond extends this principle in support of open source security software, saying, " Any security software design that doesn't assume the enemy possesses the source code is already untrustworthy ; therefore, never trust closed source.

Any collection of subsets of a set X satisfying these properties forms a base for the closed sets of a topology on X.

According to the historian of science Norwood Russell Hanson: There is no bilaterally-symmetrical, nor excentrically-periodic curve used in any branch of astrophysics or observational astronomy which could not be smoothly plotted as the resultant motion of a point turning within a constellation of epicycles, finite in number, revolving around a fixed deferent. Any path — periodic or not, closed or open — can be represented with an infinite number of epicycles.

Any of these will result in some power to the solenoid, but not enough to hold the heavy contacts closed, so the starter motor itself never spins, and the engine does not start.

" Robinson also returned to the stage in 1993 with a Broadway production of Frank Gilroy's Any Given Day, but the play closed after only six weeks.

Any two closed curves in space, if allowed to pass through themselves but not each other, can be moved into exactly one of the following standard positions.

Any small perturbation from the closed trajectory would cause the system to return to the limit cycle, making the system stick to the limit cycle.

Any compact orientable surface and any compact surface with non-empty boundary embeds in though any closed non-orientable surface needs.

Any loose thread can be cut off with scissors or tied together to form a closed circle.

Any closed 3-manifold is the boundary of a 4-manifold.

Any finite graph has a finite ( though perhaps exponential ) number of distinct simple cycles, and if the graph is embedded into three-dimensional space then each of these cycles forms a simple closed curve.

Any closed curve within the plane bounds a disk below the plane that does not pass through any other graph feature, and any closed curve through the apex bounds a disk above the plane that does not pass through any other graph feature.

Any fixed time can be used in place of the present ( e. g., the end of one interval of an annuity ); the value obtained is zero if and only if the NPV is zero.

# Any interval of reals b has measure b − a

Any 7-note scale constructed sequentially using only two types of intervals, repeating at the octave, having a tonal center, and comprising only one tritone interval between any two scale members.

Any non-trivial idempotent a is a zero divisor ( because with neither a nor b being zero, where ).

* Any Cartesian product of intervals b and d is Lebesgue measurable, and its Lebesgue measure is ( b − a )( d − c ), the area of the corresponding rectangle.

( vi ) Any compound, mixture, or preparation which contains any quantity of any of the substances referred to in paragraphs ( b )( 31 )( i ) through ( v ) of this section.

::( b ) Any action which has the aim or effect of dispossessing them of their lands, territories or resources ;

Any member of B is in the set ker r + im u ; since for all b in B, b =

Any complex number may be written a + bi, where a and b are real numbers and i is the imaginary unit.

Any quaternion may be written as ( a, b, c, d ) = a + bi + cj + dk.

Any number b + 1 is written as " 11 < sub > b </ sub >" in base b, so 11 is trivially a palindrome in base 10.

* Any positive multiple of the repunit R < sub > n </ sub >< sup >( b )</ sup > contains at least n nonzero digits in base b.

( a ) Any firearm which has a short stock and is designed to be held and fired by the use of a single hand ; and ( b ) Any combination of parts from which a firearm described in paragraph ( a ) can be assembled.

Any deduction not found in section 67 ( b ) is a miscellaneous itemized deduction.

Any institution that wishes to a ) offer a degree and / or b ) use the term " university " must be authorized to do so under an Act of the Legislature or by the Minister of Training, Colleges and Universities under the Post-secondary Education Choice and Excellence Act, 2000.

Any non-linear function, f ( a, b ), of two variables, a and b, can be expanded as

* Midnight at the Oasis b / w Any Old Time ( single by Maria Muldaur )

# 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.

# Any non-zero real number can divide both sides.

Any given individual or organization attempting to implement this model in a real situation would be unable to comply with the three requirements.

* Any Lie group G defines an associated real Lie algebra.

Any real mode operating system ( such as Xenix ) could also be made to run using OS / 2's virtual machine capabilities, subject to certain direct hardware access limitations.

Any Dedekind-complete ordered field is isomorphic to the real numbers.

Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative.

* Any field, including fields of rational numbers, real numbers, and complex numbers.

Any non-negative real number can be represented as a base-φ numeral using only the digits 0 and 1, and avoiding the digit sequence " 11 "-this is called a standard form.

* Any polynomial ( real or complex ) is an analytic function.

Any Taylor series for this function converges not only for x close enough to x < sub > 0 </ sub > ( as in the definition ) but for all values of x ( real or complex ).

Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane.

Any destruction by the Occupying Power of real or personal property belonging individually or collectively to private persons, or to the State, or to other public authorities, or to social or cooperative organizations, is prohibited, except where such destruction is rendered absolutely necessary by military operations.

Furthermore, in the opening credits there is a mock disclaimer which reads ( in translation ): “ Any resemblance to real events, to persons dead or living, is not accidental.

Any real advantages to the system were far outweighed by the additional cost of manufacture associated with the device.

Any rational function of a real variable can be written as the sum of a polynomial function and a finite number of algebraic fractions.

Any hope of real progress by the Poles was then lost.

Any entity from the real or virtual world that lends itself to dramatic interpretation may be taken up as a subject.

* Any open subset of a finite-dimensional real, and therefore complex, vector space is a diffeological space.

Any public employee who fell under his power would be replaced by someone in whom he saw real ability instead of a mere useful recipient of a favor.