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If ƒ is invertible, the function g is unique ; in other words, there can be at most one function g satisfying this property.
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If and ƒ
If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).
* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.
If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.
If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.
If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa.
If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.
If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation.
If ƒ: A → B is a homomorphism between two algebraic structures ( such as homomorphism of groups, or a linear map between vector spaces ), then the relation ≡ defined by
* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.
If and is
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 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 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 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 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 and invertible
If G = GL < sub >*</ sub >( K ), then the set of natural numbers is a proper subset of G < sub > 0 </ sub >, since for each natural number n, there is a corresponding identity matrix of dimension n. G ( m, n ) is empty unless m = n, in which case it is the set of all nxn invertible matrices.
* If A is a square matrix ( i. e., m = n ), then A is invertible if and only if A has rank n ( that is, A has full rank ).
If R = K is a field, then a series is invertible if and only if the constant term is non-zero, i. e., if and only if the series is not divisible by X.
If λI − T is invertible then that inverse is linear ( this follows immediately from the linearity of λI − T ), and by the bounded inverse theorem is bounded.
If an operator is not injective ( so there is some nonzero x with T ( x ) = 0 ), then it is clearly not invertible.
If the change of variables is given by an invertible matrix, not necessarily orthogonal, then the coefficients λ < sub > i </ sub > can be made to be 0, 1, and − 1.
If S is allowed to be any invertible matrix then B can be made to have only 0, 1, and − 1 on the diagonal, and the number of the entries of each type ( n < sub > 0 </ sub > for 0, n < sub >+</ sub > for 1, and n < sub >−</ sub > for − 1 ) depends only on A.
If the n eigenvalues are distinct ( that is, none is equal to any of the others ), then V is invertible, implying the decomposition.
* If 2 is invertible, then and are orthogonal idempotents, called symmetrizing and anti-symmetrizing, so the algebra decomposes as a direct sum of symmetric and anti-symmetric ( Hermitian and skew Hermitian ) elements.
If A is invertible, then the factorization is unique if we require that the diagonal elements of R are positive.
If A is a subalgebra of B, then for every invertible b in B the function which takes every a in A to b < sup >− 1 </ sup > a b is an algebra homomorphism ( in case, this is called an inner automorphism of B ).
If is a unique factorization domain with field of fractions, then by Gauss's lemma is irreducible in, whether or not it is primitive ( since constant factors are invertible in ); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of.
An element of this ring is invertible if a ( 1 ) is invertible in R. If R is commutative, so is Ω ; if R is an integral domain, so is Ω.
If A has no 2-torsion ( in particular, if 2 is invertible ) then the grade involution can be used to distinguish the even and odd parts of A:
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