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real and line
Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.
Then, since every real is the limit of some Cauchy sequence of rationals, the completeness of the norm extends the linearity to the whole real line.
In particular, no subset of the points on the real line gets arbitrarily close to any real number.
Compactness generalizes many important properties of closed and bounded intervals in the real line ; that is, intervals of the form for real numbers and.
For instance, any continuous function defined on a compact space into an ordered set ( with the order topology ) such as the real line is bounded.
* Consider the set of all functions from the real number line to the closed unit interval, and define a topology on so that a sequence in converges towards if and only if converges towards for all.
For example, the real line equipped with the discrete topology is closed and bounded but not compact, as the collection of all singleton points of the space is an open cover which admits no finite subcover.
is in C. In other words, every point on the line segment connecting x and y is in C. This implies that a convex set in a real or complex topological vector space is path-connected, thus connected.
Consider the real line with its ordinary topology.
It is possible to turn the real line into a compact space by adding a single " point at infinity " which we will denote by ∞.
Every sequence that ran off to infinity in the real line will then converge to ∞ in this compactification.
Intuitively, the process can be pictured as follows: first shrink the real line to the open interval (- π, π ) on the x-axis ; then bend the ends of this interval upwards ( in positive y-direction ) and move them towards each other, until you get a circle with one point ( the topmost one ) missing.
Identify each such point θ on the circle with the corresponding point on the real line tan ( θ / 2 ).
Since tangents and inverse tangents are both continuous, our identification function is a homeomorphism between the real line and the unit circle without ∞.
What we have constructed is called the Alexandroff one-point compactification of the real line, discussed in more generality below.
It is also possible to compactify the real line by adding two points, +∞ and -∞; this results in the extended real line.
It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length | r | can be constructed with compass and straightedge.

real and R
In the cases of the rational numbers ( Q ) and the real numbers ( R ) there are no nontrivial field automorphisms.
For example, the field extension R / Q, that is the field of real numbers as an extension of the field of rational numbers, is transcendental, while the field extensions C / R and Q (√ 2 )/ Q are algebraic, where C is the field of complex numbers.
NEEP, itself a joint venture, is held by DiBenedetto AS Roma LLC ( later renamed to AS Roma SPV, LLC ) and Unicredit in 60-40 ratio, which the former had 4 real person shareholders in equal ratio, led by Roma current president Thomas R. DiBenedetto.
* The set R of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover.
is a function defined on a subset I of the set R of real numbers.
This subset I is referred to as the domain of f. Possible choices include I = R, the whole set of real numbers, an open interval
The convex subsets of R ( the set of real numbers ) are simply the intervals of R.
* The cofinality of the real numbers with their usual ordering is ℵ < sub > 0 </ sub >, since N is cofinal in R. The usual ordering of R is not order isomorphic to c, the cardinality of the real numbers, which has cofinality strictly greater than ℵ < sub > 0 </ sub >.
Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.
The space R of real numbers and the space C of complex numbers ( with the metric given by the absolute value ) are complete, and so is Euclidean space R < sup > n </ sup >, with the usual distance metric.
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G ( n ) is the real number described by the nth description in the sequence.
For example, if u < sub > 1 </ sub > is an eigenvector of A, with a real eigenvalue smaller than one, then the straight lines given by the points along α u < sub > 1 </ sub >, with α ∈ R, is an invariant curve of the map.
* If x < sub > 0 </ sub > is a real number, we can turn the set R
Let R denote the field of real numbers.
For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted R < sup > n </ sup > and sometimes called real coordinate space.

real and with
But with Bill O'Connor on the fiddle, and Gran Harrow exuberantly shouting `` Glory Be '' and `` Hallelujah '' above their united chant of the lilting old ballads, they played their quaint folk games with all the fervor and abandon of a real celebration.
Thus the transformation of Adam Smith's ideal entrepreneur into a mythological detective coincides closely with the decline of the real entrepreneur in economic life.
To perpetuate wealth control led by small groups of individuals who played no role in its creation prevents those with real initiative from coming to the fore, and is basically anti-democratic.
The myth of the Southern plantation has had only a tangential relation with actuality, as Francis Pendleton Gaines showed forty years ago, and I suspect it has had a far narrower acceptance as something real than has generally been supposed.
Mercer's Whiteman association brought him into contact with Hoagy Carmichael, whose `` Snowball '' Mercer relyriced as `` Lazybones '', in which form it became a hit and marked the real beginning of Mercer's song-writing career.
It cannot become the source of a real Atlantic community if it remains organized to deal only with the military threat which first brought it into being ''.
In the interim between now and next year, we trust the House and Senate will put their minds to studying Georgia's very real economic, fiscal and social problems and come up with answers without all the political heroics.
To these people, solidarity and unity with China should be the real basis of Russia's future policy.
It seems to me that N.C., in his editorial `` Confrontation '' ( SR, Mar. 25 ), has hit upon the real problem that bothers all of us in a complex world: how do we retain our personal relationship with those who suffer??
When he was in the war, he was in Law or Supplies or something like that, and an old buddy of his told me he would come down on Sundays to the Pentagon and read the citations for medals -- just like the one we sent in for Trig -- and go away with a real glow.
one who is thoroughly human, who affects no dignity, and who is endowed with real ability, genuine worth, and sterling honesty -- all dedicated to secure the best interests of the country he has loved and served so long.
Assessors in Rhode Island are charged not only with placing a valuation upon real and personal property, but they also have the responsibility to raise by a tax `` a sum not less than nor more than '' a specified amount as ordered by a city council or financial town meeting.
This section prevents the military departments and the Office of Civil and Defense Mobilization from carrying out certain transactions involving real property unless they come into agreement with the Committees on Armed Services of the Senate and the House of Representatives.
After well broken and equipped with 12-oz. shoes on behind, bare-footed in front, she would trot a real storm with the master, Delvin, driving.
Consultation with architects, clients, real estate men, fabric houses and furniture companies is essential to the proper development of class problems just as in actual work.
These are real analytic periodic functions with period T.
We have a brief glimpse of the Tsar's public personality, the `` official Boris '', but our real focus is on the excitement of the crowd -- a significant contrast with its halfhearted acclamation in the opening scene, its bitter resentment and fury in the final act.
It is the similarity between Estella's hands and Molly's ( `` The action of her fingers was like the action of knitting '' ) that provides Pip with a vital clue to the real identity of both and establishes a symbolic connection between the underworld of crime and the genteel cruelty of Satis House.
Mr. Clark believes in a good full breakfast of fruit, hot cereal, milk, honey, whole-wheat toast with real butter and eggs.
Authorities hesitate to quote exact figures, however, believing that any sum they come up with is only a surface manifestation -- turned up by their inevitably limited policing -- of the real loot of the medical racketeer.
Nevertheless, with a reading public that longs for the `` good old days '' and with an awareness of our expanding international interests, it is easy for the Benets to obtain a magnified position in literature by use of all sorts of Americana, real or fake, and it is easy for the Steinbecks and Sandburgs to support their messages of reform by reading messages of reform into the minds of the folk.
True, it is no longer cricket for the butler to be the killer in mystery fiction, but we are dealing here with actual people in real life and not imaginary characters and situations.

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