If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

If Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

If Af is the null space of Af, then Theorem 12 says that Af.

If the argument is accepted as essentially sound up to this point, it remains for us to consider whether the patient's difficulties in orienting himself spatially and in locating objects in space with the sense of touch can be explained by his defective visual condition.

If, on the other hand, they opted for representation, it had to be representation per se -- representation as image pure and simple, without connotations ( at least, without more than schematic ones ) of the three-dimensional space in which the objects represented originally existed.

If a child loses a molar at the age of two, the adjoining teeth may shift toward the empty space, thus narrowing the place intended for the permanent ones and producing a jumble.

If elements in the sample space increase arithmetically, when placed in some order, then the median and arithmetic average are equal.

If antimatter-dominated regions of space existed, the gamma rays produced in annihilation reactions along the boundary between matter and antimatter regions would be detectable.

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 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.

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 norm of a Banach space satisfies this identity, the associated inner product which makes it into a Hilbert space is given by the polarization identity.

If X is a real Banach space, then the polarization identity is

If this suggestion is correct, the beginning of the world happened a little before the beginning of space and time.

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 you have a full-time doctor now, can he be replaced with a part-time doctor or one who serves on a fee-per-case basis only??

If the volume is the molal volume, then Af is obtained on a molal basis which is the customary terminology of the chemists.

If you are going to farm for extra cash income on a part-time basis you must keep in mind the needed machinery investments when you choose among farm enterprises.

If we use a cylindrical coordinate system with basis vectors, then the gradient of and the divergence of are given by

If is an outward pointing in-plane normal, whereas is the unit vector perpendicular to the plane ( see caption at right ), then the orientation of C is chosen so that a tangent vector to C is positively oriented if and only if forms a positively oriented basis for R < sup > 3 </ sup > ( right-hand rule ).

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

If the matrix entries are real numbers, the matrix can be used to represent two linear mappings: one that maps the standard basis vectors to the rows of, and one that maps them to the columns of.

If his son were guilty, he would still believe him innocent, on the basis of faith in his son ; this would violate the third subjunctive condition.

If " hate speech " is taken to mean ethnic agitation, it is prohibited in Finland and defined in the section 11 of the penal code, War crimes and crimes against humanity, as publishing data, an opinion or other statement that threatens or insults a group on basis of race, nationality, ethnicity, religion or conviction, sexual orientation, disability, or any comparable basis.

If we should say that no one could possibly know whether water exists a priori, it seems either we cannot know content externalism to be true on the basis of thought experiments or we cannot know what we are thinking without first looking into the world to see what it is like.

If it lapses, the legal basis for enforcing discipline disappears, and soldiers lose their legal indemnity for acts committed under orders.

" If, in turn, the speaker responds positively to the idea of giving to the poor, then some aspect of that idea must have inspired a positive response ; one could argue that that aspect is also the basis of its goodness.

If the former occurs, then any subsequent measurement performed by Bob, in the same basis, will always return 1.

If an index transforms like a vector with the inverse of the basis transformation, it is called contravariant and is traditionally denoted with an upper index, while an index that transforms with the basis transformation itself is called covariant and is denoted with a lower index.

If the employee made after-tax contributions to the non-Roth 401k account, these amounts are commingled with the pre-tax funds and simply add to the non-Roth 401 ( k ) basis.

If we fail to justify simplicity considerations on the basis of the context in which we make use of them, we may have no non-circular justification: " just as the question ' why be rational?

Humphry Davy, an English chemist was among the first to expound on the chemical basis of soil fertility and wrote in 1813: “ If land be unproductive, and a system of ameliorating it is to be attempted, the sure method of obtaining the object is to determine the cause of the sterility, which must necessarily depend upon some defect in the constitution of the soil, which may be easily discovered by chemical analysis .” His ideas were not immediately accepted.

If basis vectors are defined such that

If Naburimannu was the originator of System A, then we can on that basis place him in Babylonia sometime between the Persian and Macedonian conquests.

* If R is a Noetherian ring, then R is Noetherian by the Hilbert basis theorem.

If the president is outside of the country, or the position is vacant, the President of the Bundesrat ( a position that is rotated among the state premiers on an annual basis ) temporarily assumes the powers of the president until a successor is elected without assuming the office of president as such.

If the kernel used is a Gaussian radial basis function, the corresponding feature space is a Hilbert space of infinite dimensions.

If they are worthy rivals to each other, they also seem to make effective partners — but only on a temporary basis.