If X is a Banach space and C ⊂ X a convex subset, for x** ∈ X** and A ⊂ X** let d(x**,C) = inf||x**-x||: x ∈ C be the distance from x** to C and d̂(A,C) = supd(a,C): a ∈ A. Among other things, we prove that if X is an order-continuous Banach lattice and K is a w*-compact subset of X** we have: (i) $d̂({\overline{co}}^{w*}\left(K\right),X)\le 2d̂(K,X)$ and, if K ∩ X is w*-dense in K, then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$; (ii) if X fails to have a copy of ℓ₁(ℵ₁), then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$; (iii) if X has a 1-symmetric basis, then $d̂({\overline{co}}^{w*}\left(K\right),X)=d̂(K,X)$.

We discuss recent developments in the study of the homotopy classes for the Sobolev spaces $W^{1,p}\left(M;N\right)$. In particular, we report on the work of H. Brezis - Y. Li [5] and F.B. Hang - F.H. Lin [9].

In this article, we prove the first mean value theorem for integrals [16]. The formalization of various theorems about the properties of the Lebesgue integral is also presented.MML identifier: MESFUNC7, version: 7.8.09 4.97.1001

We consider a Banach space, which comes naturally from $c_0$ and it appears in the literature, and we prove that this space has the fixed point property for non-expansive mappings defined on weakly compact, convex sets.

In this paper, we give necessary and sufficient conditions for a point in a Musielak-Orlicz sequence space equipped with the Orlicz norm to be an H-point. We give necessary and sufficient conditions for a Musielak-Orlicz sequence space equipped with the Orlicz norm to have the Kadec-Klee property, the uniform Kadec-Klee property and to be nearly uniformly convex. We show that a Musielak-Orlicz sequence space equipped with the Orlicz norm has the fixed point property if and only if it is reflexive....

We give a simple proof of the result that if D is a (not necessarily bounded) hyperbolic convex domain in ${ℂ}^n$ then the set V of fixed points of a holomorphic map f:D → D is a connected complex submanifold of D; if V is not empty, V is a holomorphic retract of D. Moreover, we extend these results to the case of convex domains in a locally convex Hausdorff vector space.

We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on ${ℝ}^n$ for the relativistic Schrödinger operator $\sqrt{-\Delta }+Q$ to be bounded as an operator from the Sobolev space $W_2^{1/2}\left({ℝ}^n\right)$ to its dual $W_2^{-1/2}\left({ℝ}^n\right)$.

The aim of this paper is to derive by elementary means a theorem on the representation of certain distributions in the form of a Fourier integral. The approach chosen was found suitable especially for students of post-graduate courses at technical universities, where it is in some situations necessary to restrict a little the extent of the mathematical theory when concentrating on a technical problem.