* If numbers have mean X, then.

* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

If X and Y are Banach spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B ( X, Y ).

If X is a Banach space and K is the underlying field ( either the real or the complex numbers ), then K is itself a Banach space ( using the absolute value as norm ) and we can define the continuous dual space as X ′ = B ( X, K ), the space of continuous linear maps into K.

* Theorem If X is a normed space, then X ′ is a Banach space.

If X ′ is separable, then X is separable.

If F is also surjective, then the Banach space X is called reflexive.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in the weak topology.

If there is a bounded linear operator from X onto Y, then Y is reflexive.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.

If a relation is Euclidean and reflexive, it is also symmetric and transitive.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space.

If implemented using remainders of Euclidean division rather than subtractions, Euclid's algorithm computes the GCD of large numbers efficiently: it never requires more division steps than five times the number of digits ( base 10 ) of the smaller integer.

If R is a Euclidean domain in which euclidean division is given algorithmically ( as is the case for instance when R = F where F is a field, or when R is the ring of Gaussian integers ), then greatest common divisors can be computed using a form of the Euclidean algorithm based on the division procedure.

If not stated otherwise, this article considers symmetry groups in Euclidean geometry, but the concept may also be studied in wider contexts ; see below.

If X and Y are subsets of the real numbers, d < sub > 1 </ sub > and d < sub > 2 </ sub > can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, | x − y | < δ implies | f ( x ) − f ( y )| < ε.

If the Euclidean space is equipped with a choice of origin, then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin.

If the original space is Euclidean, the higher dimensional space is a real projective space.

If the starting number is rational then this process exactly parallels the Euclidean algorithm.

If the range of each chart is the n-dimensional Euclidean space, then M is said to be an n-dimensional manifold.

If we consider that its length is actually the distance from its tail to its tip, it becomes clear that the Euclidean norm of a vector is just a special case of Euclidean distance: the Euclidean distance between its tail and its tip.

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open set centered at x which is contained in S.

* If X is the Euclidean space of real numbers, then int ( 1 ) = ( 0, 1 ).

If the geometry of the universe is not compact, then it is infinite in extent with infinite paths of constant direction that, generally do not return and the space has no definable volume, such as the Euclidean plane.

If working over a ring where SL is generated by transvections ( such as a field or Euclidean domain ), one can give a presentation of SL using transvections with some relations.

* If π < sub > 1 </ sub >( M ) is virtually abelian but not virtually cyclic then the geometric structure on M is Euclidean, and M is compact.

If we specifically choose the Euclidean norm on both R < sup > n </ sup > and R < sup > m </ sup >, then we obtain the matrix norm which to a given matrix A assigns the square root of the largest eigenvalue of the matrix A < sup >*</ sup > A ( where A < sup >*</ sup > denotes the conjugate transpose of A ).

* If deg P deg Q, then it is necessary to perform the Euclidean division of P by Q, using polynomial long division, giving P ( x )

* If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged.

If the process depends only on | x-x '|, the Euclidean distance ( not the direction ) between x and x then the process is considered isotropic.

If X is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.