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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: S2n-1→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 Fourier-Lebesgue 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 non-Abelian group [Ref.
      
It is also proved that the adjoint operator is an RN-operator if and only if for every separable subspace Xo of X the set (T|Xo)*(Y*) is separable (Theorem 2).
      
It is proved that a WCG-space E is conjugate to a Banach space if and only if its conjugate space E' contains a norm-closed 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 m-admissible 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 theA-affine 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 asymptoticallyCr-regular,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 ofp-norms of its μ-means 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 self-dual is proved and applied to obtain new proofs of earlier results.
      
Namely, we prove that the Saks-Henstock lemma is valid in a Banach space if and only if it is finite-dimensional.
      
 

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