# Blog Posts

Vector spaces over a field are a special case of the more general notion of modules over a ring. Rather than the long list of axioms normally presented in textbooks, we see how an algebraic view of vector spaces helps to motivate the definition of modules.

A summary of basic concepts in topology, including topological spaces, compactness, separation, convergence, and continuity.

The

**Chebyshev polynomials**appear frequently in numerical analysis and are incredibly useful for analyzing and accelerating the convergence of iterative methods. One might even say that Chebyshev polynomials are the best polynomials, a fact which can be made precise in a variety of different ways. In these notes, we define Chebyshev polynomials and their basic properties, before discussing their utility in minimax approximation theory, which was the subject of a previous set of notes.First formalized by Weierstrass in 1885, approximation theory concerns the best approximation of arbitrary functions by some class of simpler functions. The structure of an approximation problem involves three central components: a function class containing the function to be approximated, a form of approximating function, and a norm for measuring approximation error. These notes introduce

**minimax polynomial approximation**, whereby continuous functions on a closed interval are approximated by polynomials using the infinity norm to measure fit.