* 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 G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed.

If X is a topological space, there is a natural way of transforming X /~ into a topological space ; see quotient space for the details.

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open ( X ) under inclusion.

* If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application defined by is continuous.

If the first-countability axiom were imposed on the topological spaces in question, the two above conditions would be equivalent.

If X is a topological space, a net in X is a function from some directed set A to X.

If ( x < sub > α </ sub >) is a net in the topological space X, and x is an element of X, we say that the net converges towards x or has limit x and write

If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous, so that F is a topological field.

If X is equipped with the weak topology, then addition and scalar multiplication remain continuous operations, and X is a locally convex topological vector space.

If X and Y are topological vector spaces, the space L ( X, Y ) of continuous linear operators f: X → Y may carry a variety of different possible topologies.

If H is a subgroup of G, the set of left or right cosets G / H is a topological space when given the quotient topology ( the finest topology on G / H which makes the natural projection q: G → G / H continuous ).

* If X is a topological space, then the category of all ( real or complex ) vector bundles on X is not usually an abelian category, as there can be monomorphisms that are not kernels.

* If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category.

If the topology of the topological vector space is induced by a metric which is homogenous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

* If is a topological space ( viewed as a category as above ) and is some small category, we can form the category of all contravariant functors from to, using natural transformations as morphisms.

If one does not require a ring to have a unit, then one has to add the requirement of continuity of the additive inverse, or equivalently, to define the topological ring as a ring which is a topological group ( for +) in which multiplication is continuous, too.

If it is not, then it can be completed: one can find an essentially unique complete topological ring S which contains R as a dense subring such that the given topology on R equals the subspace topology arising from S.

If V is some topological space, for example a subset of some R < sup > n </ sup >, real-or complex-valued continuous functions on V form a commutative ring.

If one bends and deforms the surface, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will.

If a topological space is a Baire space then it is " large ", meaning that it is not a countable union of negligible subsets.