* 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 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 the descent were truly unbroken, father-to-son, since Confucius's lifetime, the males in the family would all have the same Y chromosome as their direct male ancestor, with slight mutations due to the passage of time.

If, i. e., it has a large norm with each value of s, and if, then Y ( s ) is approximately equal to R ( s ) and the output closely tracks the reference input.

If ( x, y ) are the Cartesian coordinates of a point, then (− x, y ) are the coordinates of its reflection across the second coordinate axis ( the Y axis ), as if that line were a mirror.

If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:

If f: X → Y morphism of pointed spaces, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.

If f: X < sub > 1 </ sub > → X < sub > 2 </ sub > and g: Y < sub > 1 </ sub > → Y < sub > 2 </ sub > are morphisms in C, then the group homomorphism Hom ( f, g ): Hom ( X < sub > 2 </ sub >, Y < sub > 1 </ sub >) → Hom ( X < sub > 1 </ sub >, Y < sub > 2 </ sub >) is given by φ g o φ o f.

If f: X → Y is a continuous map, x < sub > 0 </ sub > ∈ X and y < sub > 0 </ sub > ∈ Y with f ( x < sub > 0 </ sub >) = y < sub > 0 </ sub >, then every loop in X with base point x < sub > 0 </ sub > can be composed with f to yield a loop in Y with base point y < sub > 0 </ sub >.

If such a function exists, we say X and Y are homeomorphic.

If it were not for an old professor who made me read the classics I would have been stymied on what to do, and now I understand why they are classics ; ;

If I even hint at it do you think it will matter that you are his nephew -- and not even a blood nephew ''??

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

Defoe then commented, `` If they Could Draw that young Gentleman into Their Measures They would show themselves quickly, for they are not asham'd to Say They want only a head to Make a beginning ''.

If Jews are identified as a religious body in a controversy that comes before a national or international tribunal, it is obviously compatible with the goal of human dignity to protect freedom of worship.

If they are right, they will prevail of and by themselves.

Without preliminaries, Esther asked him, `` If you are a world citizen, will you take Garry Davis' place in his tent while he goes to the hospital ''??

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

If the would-be joiner asks these questions he is not likely to be duped by extremists who are seeking to capitalize on the confusions and the patriotic apprehensions of Americans in a troubled time.

If we break the minister to our bit, we are buying back our own sins.

If the record buyer's tastes are somewhat eclectic or even the slightest bit esoteric, he will find them satisfied on educational records.

If Daddy's books are out of bounds his own picture books are not.

If it will simply delay the debates until the qualifications are closed next spring, and then carry all the candidates on a tour of debates, it can provide a service to the state.

If they are to be commended for foresight in their planning, what then is the judgment of a town council that compounds this problem during the planning stage??

If the Communists are sincere in wanting a united, neutral and disarmed Germany, it might well be advantageous for the German people in this nuclear age.

If only this could be done more often -- with such heartening results -- many of the earth's `` big problems '' would shrink to the insignificances they really are.

If any are left, presently, we may expect to see signs specifically prohibiting the feeding of them too.

If the raw population figures are crucially relevant, then it is idle to think of liberation, as idle as to suppose that Poland might liberate Russia.

If it is not enough that all of our internationalist One Worlders are advocating that we join this market, I refer you to an article in the New York Times' magazine section ( Nov. 12, 1961 ), by Mr. Eric Johnston, entitled `` We Must Join The Common Market ''.

If the UN troops are not mercenaries then the Hessians were not mercenaries either.