A Fix-Finite Approximation Theorem

Fix Point Theorem for Compact Spaces

The articles [10], [11], [1], [5], [8], [7], [12], [3], [9], [4], [2], and [6] provide the notation and terminology for this paper. In this paper M denotes a non empty metric space. One can prove the following proposition (1) Let F be a set. Suppose F is finite and F 6= / 0 and F is ⊆-linear. Then there exists a set m such that m ∈ F and for every set C such that C ∈ F holds m⊆C. Let M be a non...

Fuzzy Best Simultaneous Approximation of a Finite Numbers of Functions

Fuzzy best simultaneous approximation of a finite number of functions is considered. For this purpose, a fuzzy norm on $Cleft (X, Y right )$ and its fuzzy dual space and also the  set of subgradients of a fuzzy norm are introduced. Necessary case of a proved theorem about characterization of simultaneous approximation will be extended to the fuzzy case.

.1 Kronecker's Approximation Theorem

Theorem 2 (Kronecker Approximation Theorem (1884)) ∃x ∈ Z s.t. Ax = b if and only if ∀y, y>b is an integer whenever y>A is an integral vector. Proof: To prove the forward implication, take an integral solution x∗. For any y, since y>Ax∗ = y>b, if y>A is integral then y>b must be an integer too. We next prove the converse. First, suppose that Ax = b does not have a solution (over the reals). The...

The Block Approximation Theorem

The block approximation theorem is an extensive generalization of both the well known weak approximation theorem from valuation theory and the density property of global fields in their henselizations. It guarantees the existence of rational points of smooth affine varieties that solve approximation problems of local-global type (see e.g. [HJP07]). The theorem holds for pseudo real closed field...

Analytic Approximation Theorem