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 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 we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

If and we have for all v, w in V, then we say that B is symmetric.

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

If is a ket in V and is a ket in W, the direct product of the two kets is a ket in.

If an epidemic of cholera is suspected, the most common causative agent is V. cholerae O1.

If V. cholerae serogroup O1 is not isolated, the laboratory should test for V. cholerae O139.

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

If the electrode has a positive potential with respect to the SHE, then that means it is a strongly reducing electrode which forces the SHE to be the anode ( an example is Cu in aqueous CuSO < sub > 4 </ sub > with a standard electrode potential of 0. 337 V ).

* 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 V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

# If V is a set of strings then V * is defined as the smallest superset of V that contains λ ( the empty string ) and is closed under the string concatenation operation.

# If V is a set of symbols or characters then V * is the set of all strings over symbols in V, including the empty string.

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 he had been `` liquidated '' in some way, he would have become a martyr, a rallying point for people who shared his ideas.

If I now risk some comparisons with Sons And Lovers let it be clear that I am not comparing the two works or judging their merits ; ;

If, as Reid says, `` nearly all his poetry was produced when he was not taking opium '', there may be some reason to doubt that he was under its influence in the period from 1896 to 1900 when he was writing the poems to Katie King and making plans for another book of verse.

If he can't play with Mommy's magazines, he should have some old numbers of his own.

If some future Khrushchev decided to rake up the misdeeds of his revered predecessor, would not the factory workers pass the same resolutions applauding his dispossession??

If Nikita buys a small plot in some modest rural cemetery, everyone will understand.

If he were to go with White, he would be out there two days, not just listening in the dark at some point between here and Papa-san, but moving ever deeper into enemy land -- behind Papa-san -- itself.

If I don't get some whisky down him he might die.

If those aren't enough for you, why not grow some just for winter blooming??

If, for some reason such as economy, we are not going to develop aircraft nuclear propulsion with a sense of national urgency, then we should turn our effort to developing jet engines with a thrust-to-weight ratio of 12 or 15 to one.

If the dance teachers of America make it their business to prepare their young charges for the gymnastics that must come some day if our schools are really responsible, we will be that much ahead.

If this happens, there may be some class conflict in the South, with school boards and school teachers taking the middle-class position.

If this threefold division of costs were to have its counterpart in the actual rates of charge for service, as it actually does have in some rates, there would result a three-part rate for any one class of service.

If the photographically realistic continuity of dreams, however bizarre their combinations, denies that it is purely a composition of the brain, it must be compounded from views of diverse realities, although some of them may never be encountered in what we are pleased to call the real life.

If Depew had told any academic psychologist that he had a weird feeling of having lived through that identical convention session at some time in the past, he would have been informed that he was a victim of deja vue.

If asked how she gets her impressions, she probably can only say that she `` just gets them '' -- some more vividly than others.

If you want to raise feed or carry out some enterprise on a larger scale, you'll need more land.

If you draw the short straw I'll lend you some bread, like fifty bucks, before I take off to visit my sister in Frisco.

If a substitute mechanism is needed for the control of a fictitious impersonal market, quite obviously some method must be devised for representing the public interest.

There are some passages in the writings of Irenaeus where the image of God and the similitude are sharply distinguished, so most notably in the statement: `` If the ( Holy ) Spirit is absent from the soul, such a man is indeed of an animal nature ; ;

I kept saying, `` If I could just build up a reputation for myself, make some real money, get to be well known as an illustrator -- like Peter Askington, for instance -- then I could take some time off and paint ''.

If she and Pete were really getting serious, I'd have to do some hard thinking.

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

If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.