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

If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.

* If an R module is finitely generated and projective ( that is, a progenerator ), then the endomorphism ring of the module and R share all Morita invariant properties.

If G arises as the group of units of a ring A, then an inner automorphism on G can be extended to a projectivity on the projective space over A by inversive ring geometry.

#( Betti numbers ) If X is a ( good ) " reduction mod p " of a non-singular projective variety Y defined over a number field embedded in the field of complex numbers, then the degree of P < sub > i </ sub > is the i < sup > th </ sup > Betti number of the space of complex points of Y.

If D is finite then this constructs a finite projective space.

If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set.

If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

If the ring has special properties, this hierarchy may collapse, i. e. for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well.

It is shown, however, that this leads to an extension problem for G. If G is correctly extended we can speak of a linear representation of the extended group, which gives back the initial projective representation on factoring by F < sup >∗</ sup > and the extending subgroup.

* If e = e < sup > 2 </ sup > is an idempotent in the ring R, then Re is a projective left module over R.

The length of a finite resolution is the subscript n such that P < sub > n </ sub > is nonzero and P < sub > i </ sub >= 0 for i greater than n. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its projective dimension and denoted pd ( M ).

If M does not admit a finite projective resolution, then by convention the projective dimension is said to be infinite.

( If a quotient module R / I, for any commutative ring R and ideal I, is a projective R-module then I is principal.

If B is a ( possibly non-commutative ) A-algebra that is a finitely generated projective A-module containing A as a subring, then A is a direct factor of B.

If there is some notion of " localization " which can be carried over to modules, such as is given at localization of a ring, one can define locally free modules, and the projective modules then typically coincide with the locally free ones.

If one wants to consider antipodal points as identified, one passes to projective space ( see also projective Hilbert space, for this idea as applied in quantum mechanics ).

To take something more interesting, let V be the projective line over F. If F has q elements, then this has q + 1 points, including as we must the one point at infinity.

If real or complex numbers are used, then, from the point of view of differential geometry, points at infinity form a hypersurface, which means a submanifold having one less dimension than the whole projective space.

If the affine 3-space is real,, then the addition of a real projective plane at infinity produces the real projective 3-space.