Related papers: A short and elementary proof of Hanner's theorem

The Taylor expansion is a widely used and powerful tool in all branches of Mathematics, both pure and applied. In Probability and Mathematical Statistics, however, a stronger version of Taylor's classical theorem is often needed, but only…

While exploring dynamical systems, we often come across the principle of contraction mapping, or better known as the Banach fixed point theorem. It is an essential concept based on successive approximation, whose utility comes from two main…

A proof based on the Chern-Gauss-Bonnet Theorem is given to Hopf Theorem concerning the degree of the Gauss map of a hypersurface in $\mathbb{R}^n$.

We present a concise proof for the supporting hyperplane theorem. We then observe that the proof not only establishes the supporting hyperplane theorem but also extends it to a hyperplane separation theorem for certain non-convex sets. The…

We present a simple short proof of the Fundamental Theorem of Algebra, without complex analysis and with a minimal use of topology. It can be taught in a first year calculus class.

Here, we give a self-contained and elementary proof of a minimax theorem due to Fan in a simplified setting that can be taught in an advanced undergraduate course. Our proof follows Nikaido's argument with some simplifications.

A topological space is locally equiconnected if there exists a neighborhood $U$ of the diagonal in $X\times X$ and a continuous map $\lambda:U\times[0,1]\to X$ such that $\lambda(x,y,0)=x$, $\lambda(x,y,1)=y$ et $\lambda(x,x,t)=x$ for…

Rectangular TVS-cone metric spaces are introduced and Kannan's fixed point theorem is proved in these spaces. Two approaches are followed for the proof. At first we prove the theorem by a direct method using the structure of the space…

We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial.

We reprove the strong Hanani-Tutte theorem on the projective plane. In contrast to the previous proof by Pelsmajer, Schaefer and Stasi, our method is constructive and does not rely on the characterization of forbidden minors, which gives…

In this paper, two main results concerning uniformly continuous retractions are proved. First, an $\alpha$-H\"older retraction from any separable Banach space onto a compact convex subset whose closed linear span is the whole space is…

We prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being `small' in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation…

We provide a different proof of the equivariant version of the Borsuk-Whitehead-Hanner Theorem in the category of proper G-spaces which are metrizable by a G-invariant metric.

We show that if $L$ is a topological vector lattice, $u \colon L \to L$ is the function $u(x) = x \vee 0$, $C \subset L$ is convex, and $D = u(C)$ is metrizable, then $D$ is an ANR and $u|_C \colon C \to D$ is a homotopy equivalence and…

We will present a novel elementary, self-contained, and explicit proof of the local Kronecker-Weber theorem. Apart from discrete valuation theory, it does not make use of any tools beyond those introduced in a second undergraduate course on…

Let $X$ and $E$ be $f$-algebras and $p:X \to E_+$ be a monotone vector norm. Then the triple $(X,p,E)$ is called a lattice-normed $f$-algebraic space. In this paper, we show a generalization of the extension of the Hahn-Banach theorem for…

Brehm's extension theorem states that a non-expansive map on a finite subset of a Euclidean space can be extended to a piecewise-linear map on the entire space. In this note, it is verified that the proof of the theorem is constructive…

We present a constructive proof of Brouwer's fixed point theorem for uniformly continuous and sequentially locally non-constant functions based on the existence of approximate fixed points. And we will show that Brouwer's fixed point…

The Hahn-Banach theorem is an extension theorem for linear functionals which preserves certain properties. Specifically, if a linear functional is defined on a subspace of a real vector space which is dominated by a sublinear functional on…

We use discrete Morse theory to give a new proof of the Degree Theorem in Auter space A_n. There is a filtration of A_n into subspaces A_{n,k} using the degree of a graph, and the Degree Theorem says that each A_{n,k} is (k-1)-connected.…