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If V is a real vector space, then we replace V by its complexification V ⊗< sub > R </ sub > C and let g denote the induced bilinear form on V ⊗< sub > R </ sub > C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g |< sub > W </ sub > = 0.
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If and V
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 and is
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 and real
If competition beats you to it, this exciting new product era can have real headaches in store.
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.
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 one quantizes a real scalar field, then one finds that there is only one kind of annihilation operator ; therefore, real scalar fields describe neutral bosons.
Existence is the principle that gives reality to an essence not the same in any way as the existence: " If things having essences are real, and it is not of their essence to be, then the reality of these things must be found in some principle other than ( really distinct from ) their essence.
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.
If X is a real Banach space, then the polarization identity is
If A is a self-adjoint operator, then is always a real number ( not complex ).
If the Czech ethnic origin was to be stressed, combinations like " Bohemian of Bohemian language "( Čech českého jazyka ), " a real Bohemian " ( pravý Čech ) etc.
If the exponent r is even, then the inequality is valid for all real numbers x.
If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.
If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.
If the eigenvalues are all positive, then the frequencies are all real and the stationary point is a local minimum.
If zeal for orthodoxy caused him to overstep the limits of discretion, his real attitude towards Rome is sufficiently clear.
If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles at complex plane origin ( i. e. their real and complex component is zero in the continuous time case ).
If this limit exists, then it may be computed by taking the limit as h → 0 along the real axis or imaginary axis ; in either case it should give the same result.
If x and y are real numbers, and if the graph of y is plotted against x, the derivative measures the slope of this graph at each point.
If a DBMS system responses users ' request in a given time period, it can be regarded as a real time database.
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 this operation has to be done in real time video games there is an easy trick to boost performance.
* If we think of as the set of real numbers, then the direct product is precisely just the cartesian product,.
* If we think of as the group of real numbers under addition, then the direct product still consists of.
* If we think of as the ring of real numbers, then the direct product again consists of.
The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring.
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