If V is finite-dimensional, then V * has the same dimension as V. Given a basis

Given a left neutral element and for any given then A4 ’ says there exists an such that.

Given also a measure on set, then, sometimes also denoted or, has as its vectors equivalence classes of measurable functions whose absolute value's-th power has finite integral, that is, functions for which one has

Given a trigonometric series f ( x ) with S as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that had S ' as its set of zeros, where S ' is the set of limit points of S. If p ( 1 ) is the set of limit points of S, then he could construct a trigonometric series whose zeros are p ( 1 ).

Given that Gable and Cukor had worked together before, in Manhattan Melodrama and Gable had no objection to working with him then, and given Selznick's desperation to get Gable for Rhett Butler, if Gable had any objections to Cukor, certainly they would have been expressed before he signed his contract for the film.

Given the proposition that ' a persons life prospects should not be decided by factors outside of their control or for which a person cannot claim personal credit ' ( i. e. social status, inherited wealth, race and other accidents of birth ) a meritocracy proposes a system where people are rewarded based on their efforts, and if everyone can start on equal footing with the same opportunity to advance, then the results are just.

Given Poseidon's connection with horses as well as the sea, and the landlocked situation of the likely Indo-European homeland, Nobuo Komita has proposed that Poseidon was originally an aristocratic Indo-European horse-god who was then assimilated to Near Eastern aquatic deities when the basis of the Greek livelihood shifted from the land to the sea, or a god of fresh waters who was assigned a secondary role as god of the sea, where he overwhelmed the original Aegean sea deities such as Proteus and Nereus.

Given a functor U and an object X as above, there may or may not exist an initial morphism from X to U. If, however, an initial morphism ( A, φ ) does exist then it is essentially unique.

Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

# Given any elements a and b of G, if a ~ b, then b ~ a ( symmetry );

# Given any elements a, b, and c of G, if a ~ b and b ~ c, then a ~ c ( transitivity );

# Given any elements a, a ', b, and b ' of G, if a ~ a ' and b ~ b ', then a * b ~ a ' * b ' ;

# Given any elements a and a ' of G, if a ~ a ', then a < sup >− 1 </ sup > ~ a ' < sup >− 1 </ sup > ( this can actually be proven from the other four, so is strictly redundant ).

Given an embedding of a rooted tree in the plane, if one fixes a direction of children ( starting from root, then first child, second child, etc.

Given " mappings " of input variables into membership functions and truth values, the microcontroller then makes decisions for what action to take, based on a set of " rules ", each of the form:

: Given any set A and any set B, if for every set C, C is a member of A if and only if C is a member of B, then A is equal to B.

: Given any set A and any set B, if A is a nonempty set ( that is, if there exists a member C of A ), then if A and B have precisely the same members, then they are equal.

Given the type of warfare that had been carried out in the conflict — wherein highly mobile groups of soldiers and rebels continued to move to and from on the desert terrain, retreating at a time and then suddenly attacking to regain control of small towns and villages in the Eastern rebel held parts of Libya — had led to the technical becoming a vehicle of choice for both sides.

# Given u and v in W, then they can be expressed as u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) and v = ( v < sub > 1 </ sub >, v < sub > 2 </ sub >, 0 ).

# Given u in W and a scalar c in R, if u = ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, 0 ) again, then cu = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, c0 ) = ( cu < sub > 1 </ sub >, cu < sub > 2 </ sub >, 0 ).

Given subspaces U and W of a vector space V, then their intersection U ∩ W :=

Given any curve C and a point P on it, there is a unique circle or line which most closely approximates the curve near P, the osculating circle at P. The curvature of C at P is then defined to be the curvature of that circle or line.

Given an eclipse, then there is likely to be another eclipse after every period P. This remains true for a limited time, because the relation is only approximate.

( Please note that Japan has a guideline of a limit of 1 % of GDP on defense spending, however, Japan defines a number of activities as non-defense spending ) Given these circumstances, some have viewed Article 9 as increasingly irrelevant.

Given the large number of U. S. military personnel and their dependents residing in Europe, it was expected that over 7 % of donors would be deferred due to the policy.

Given this inherent distortion, it is not unusual for a television or video projector to truthfully claim a linear contrast ratio ( difference between darkest and brightest output levels ) of 1000: 1 or greater, equivalent to 10 bits of audio precision even though it may only accept signals with 8-bit precision and use an LCD panel that only represents 6 or 7 bits per channel.

Given the vast territory of the Russian Far East, 6. 7 million people translates to slightly more than one person per square kilometer, making the Russian Far East one of the most sparsely populated areas in the world.

* 7 " Promo ( Given away free with initial copies of Sweaty Handshake album, 27 January 1995 )

Given this grammar, we can now say that the number of NPs expected while deriving VPs is 0. 7 x 1 + 0. 3 x 2 = 1. 3.

Given the need for such warheads to reenter the atmosphere swiftly and retain hypersonic velocities to sea level, researchers developed what are known as SHARP materials, typically hafnium diboride and zirconium diboride, whose thermal tolerance exceeds 3600 C. SHARP equipped vehicles can fly at Mach 11 at 30 km altitude and Mach 7 at sea level.

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

Given its harbors and fortifications, Saint Thomas still retained a strategic importance, and thus, in the 1860s, during the American Civil War and its aftermath, the United States government considered buying the island and its neighbors from Denmark for $ 7. 5 million, but at the time failed to find domestic legislative support for the bid.

Given 20 minutes a game, Szczerbiak averaged 7 points, 3. 2 rebounds, and 1. 1 assists while shooting. 450 % from the field and. 411 % from 3.

Given this pressure, the collapse of Taligent, and the growing problems with the existing operating system, after System 7. 5 was released in autumn 1994, Apple management decided that the decade-old operating system had run its course.

On 1 February 2009, Given transferred to Manchester City for £ 7 million on a four-and-a-half-year contract.

On 1 February 2009 Given signed with Manchester City on a four-half-year-contract for £ 7 million.

" Given kept a clean sheet in a man of the match performance in his debut, a 1 – 0 win against Middlesbrough on 7 February.

Given the municipality ’ s weak legal position, however, the legal dispute, which went on for 7 years, can no longer be further pursued.

Given that the French were attacking and exposed to artillery fire a total for them of 7, 000 killed, wounded and prisoners does not seem unreasonable, and a total of 5, 000 casualties for the Russians seems a good estimate.

Given recent comebacks such as The Miracle at the New Meadowlands ( where the Philadelphia Eagles won by scoring 28 points in the last 7 minutes ) and how quickly points can be accumulated ( through interception returns, onside kicks and kick returns ) it is understandable that coaches are cautious about becoming overconfident in their offenses and they normally prefer to run out the clock rather than risk an unlikely but certainly possible comeback late in the game, particularly for teams who have a strong offence but a weaker defense.

Given a surface temperature of 7, 740 Kelvin, stellar models yield a total luminosity for the star of about 17 times the luminosity of the Sun.

*" Empty At The End " / " This Given Line " ( DB Records CD / 7 " Single, May 2002 ) UK # 39

* Given the station's February 2008 format adjustment, it is not yet known whether WBOS will continue with their " Studio 7 " broadcasts.

Given the size of the city, with nearly 7, 000 inhabitants, there is a remarkable number of RV parks, resorts and stores.

:* The Music Tapes and Dad ( Given free with ' Stop Smiling Magazine ' # 8 ; 7 "; 2000 )

* The Music Tapes and Dad ( Given free with ' Stop Smiling Magazine ' No. 8 ; 7 "; 2000 )

*" You've Given Me Something That I Can't Give Back "-CD + 7 ", Moshi Moshi 2005