A simple proof of Sanov’s theorem*

A simple self-contained proof of Sanov's theorem in τ-topology is given, well suited for a first course on large deviations. *This work was supported by the Hungarian National Foundation for Scientific Research under Grant T046376     

A simple proof for the matrix-geometric theorem

For block-partitioned matrices of the GI/M/1 type, it has been shown by M. F. Neuts that the stationary probability vector, when it exists, has a matrix-geometric form. We present here a new proof, which we believe to be the simplest available today.     

A simple proof for Schur's theorem

In 1917 Schur gave a characterization of bounded analytic functions on the unit disc. We present a simple proof.

     

A probabilistic proof of the fundamental theorem of algebra

We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

     

A nonstandard proof of the Eberlein-Smulian theorem

The Eberlein-Smulian theorem on the equivalence of weak compactness and the finite intersection property of bounded closed convex sets is given a short elementary proof by applying Abraham Robinson's nonstandard characterization of compactness.

     

One more simple proof of the Craig–Sakamoto theorem

We give one more elementary proof of the Craig-Sakamoto’s theorem: given such that ; then AB=0.     

A simple proof of the majorizing measure theorem

We gieve a completely elementary proof of the existence of majorizing measures for bounded Gaussian processes. The proof relies upon Sudakov's minoration, the concentration of measure phenomenon, and a (somewhat deceptively) simple construction.     

A simple proof of the elliptical range theorem

A short proof is given of the elliptical range theorem concerning the numerical range of a complex matrix.

     

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.     

A simple proof of Singer's representation theorem

Let be a compact Hausdorff space and a Banach space. Singer's theorem states that under the dual pairing , the dual space of is isometric to . Using the Hahn-Banach theorem and the (scalar) Riesz representation theorem, a proof of Singer's theorem is given which appears to be simpler than the proofs supplied earlier by Singer (1957, 1959) and Dinculeanu (1959, 1967).

     

A new proof of Poltoratskii's theorem

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on and any a.e. w.r.t. μ_sing, where μ_sing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, and F_μ(z)=∫(x−z)⁻¹dμ(x).     

A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.