Let $X$ be a Banach space, $V\subset X$ is its subspace and $U\subset X^{*}$. Given $x\in X$, we are looking for $v\in V$ such that $u\left(v\right)=u\left(x\right)$ for all $u\in U$ and $\Vert v\Vert \le M\Vert x\Vert$. In this article, we study the restrictions placed on the constant $M$ as a function of $X,V$, and $U$.

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spaces and $c_0$. We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator from or $c_0$ onto anyone of their maximal proper subspaces.

The two-dimensional stationary flow of a fluid over an obstacle lying on the bottom of a stream is discussed. We take into account the gravity and we neglect the effects of the surface tension. An existence theory for the solution of this problem is established by the implicit function theorem, for small obstacles and Froude numbers in an interval included in $\left]0,1\right[$.

The concept of convexity plays an important role in the classical geometry of normed spaces and it is frequently used in several branches of nonlinear analysis. In recent years some papers that contain generalizations of the concept of convexity with the help of the measures of noncompactness have appeared. The Tauberian operators were introduced by Kalton and Wilansky (1976) and they appear in the literature with the aim of responding to some questions related with the summability and the factorization of operators; in the preservation by isomorphisms in Banach spaces, and so forth. In this paper we make the study of the Tauberian operators, not starting from the Euclidean distance, but by means of general set quantities.