On Weierstrass extreme value theorem

The Stone-weierstrass Theorem

The really new thing about Stone’s approach to the approximation theorem was the approach via lattices of continuous functions, although Lebesgue had noticed the importance of approximating the absolute-value function earlier. There is a segment of the mathematical community formed of people who are as likely to encounter a lattice in their work as an algebra and for whom the lattice version of...

The Theorem of Weierstrass

The basic purpose of this article is to prove the important Weier-strass' theorem which states that a real valued continuous function f on a topological space T assumes a maximum and a minimum value on the compact subset S of T , i.e., there exist points x1, x2 of T being elements of S, such that f(x1) and f(x2) are the supremum and the innmum, respectively, of f(S), which is the image of S und...

An Effective Weierstrass Division Theorem

We prove an effective Weierstrass Division Theorem for algebraic restricted power series with p-adic coefficients. The complexity of such power series is measured using a certain height function on the algebraic closure of the field of rational functions over Q. The paper includes a construction of this height function, following an idea of Kani. We apply the effective Weierstrass Division Theo...

Strengthened Stone-weierstrass Type Theorem

The aim of the paper is to prove that if L is a linear subspace of the space C(K) of all real-valued continuous functions defined on a nonempty compact Hausdorff space K such that min(|f |, 1) ∈ L whenever f ∈ L, then for any nonzero g ∈ L̄ (where L̄ denotes the uniform closure of L in C(K)) and for any sequence (bn)n=1 of positive numbers satisfying the relation P∞ n=1 bn = ‖g‖ there exists a se...

Generalized Central Limit Theorem and Extreme Value Statistics