

  if and only if 
We prove that a mapping T: E → F' is an extreme point of the unit ball of the space I (E, F') of integral mappings, if and only if it has the formTx=>amp;lt;x, a0>amp;gt;b0, wherea, εextS (E') andb0∈extS (F').


In this note we show that an infinitely divisible (i.d.) distribution function F is Poisson if and only if it satisfies the conditions F(+0) >amp;gt; 0, for any 0 >amp;lt; ∈ >amp;lt; 1


As topological applications, in particular, it is shown that continuous mappings of the sphere f, g: S2n1→Sn have one and the same Hopf invariant if and only if the induced chain of mappings is strongly homotopic.


Then each sequence {bn} such that bn ↓ 0, bn?αn, n=1, 2,..., is a sequence of FourierLebesgue coefficients with respect to the system {cos nx} if and only if the sequence


We know that a group G is functionally complete if and only if it is either trivial or a finite simple nonAbelian group [Ref.


It is also proved that the adjoint operator is an RNoperator if and only if for every separable subspace Xo of X the set (TXo)*(Y*) is separable (Theorem 2).


It is proved that a WCGspace E is conjugate to a Banach space if and only if its conjugate space E' contains a normclosed total subspace M, consisting of functionals which attain supremum on the unit sphere.


It is proved, in particular, that a separable normalizing subset Y of X' is madmissible for X if and only if every σ(X, Y)compact set is separable in X.


The following generalized Nikodym theorem is established: the family {μα} is uniformly bounded on Σ if and only if it is bounded on every sequence of pairwise disjoint sets of which the union is a part of some set in Σ.


Such surfaces are shown to be holomorphically equivalent if and only if they are affinely equivalent.


It is shown that theAaffine horizontal distribution onMA exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.


We prove that Γ is the Grassmannian image of a regular submanifoldFn of Euclidean spaceEn+p if and only if the curve Γ in the Grassmann manifoldG+(p, n+p) is asymptoticallyCrregular,r>amp;gt;1.


The main result is as follows: a function, belongs to if and only if for each bounded continuous function, the function is Stepanov almost periodic (of order 1) and


In the second case, it is shown that a (closed) setF can be the setJa for a certain function if and only if the projections ofF on the coordinate axes nowhere dense.


It is clear that a Lie ringR() is commutative if and only if the semigroupM(R) (orA(R)) is commutative.


It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR() is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A.


Then the trigonometric series is the Fourier series of a functionf ∈Lp(), wherep ε ]1; ∞[, if and only if the sequence ofpnorms of its μmeans is bounded: In the case of the Fejér method, we have the test due to W.


We also prove that the operatorPY, whereY ?C[K] is a nontrivial Chebyshev subspace andK is a compactum, is linear if and only if the codimension ofY inC[K] is equal to 1.


The theorem stating that a cone in a Hilbert space is regular if and only if it is selfdual is proved and applied to obtain new proofs of earlier results.


Namely, we prove that the SaksHenstock lemma is valid in a Banach space if and only if it is finitedimensional.




首页上一页12345下一页尾页


