Subespacios densos de $C[0,1]$. Teoremas de Stone-Weierstrass y de M\"untz-Sz\'asz

TL;DR

This paper explores dense subspaces of continuous functions on [0,1], focusing on algebraic structures and monomial-generated subspaces, highlighting key theorems like Stone-Weierstrass and M"untz-Sz"asz.

Contribution

It generalizes classical approximation theorems to new subspace classes, including algebraic and monomial-generated spaces, and reviews recent developments in the field.

Findings

Stone-Weierstrass theorem for algebraic subspaces

M"untz-Sz"asz theorem for monomial-generated subspaces

Recent advances in dense subspace characterization

Abstract

In this work, we look into some results about dense subspaces of $C[0,1]$. Being our starting point the Weierstrass' Approximation Theorem, we study generalization of this in two directions: the first one studying subspaces which also have algebra structure, where the main result will be the Stone-Weierstrass theorem, and the second one will be considering subspaces generated by sets of monomials whose exponents satisfy certain properties, where the central result will be the M\"untz-Sz\'asz theorem. Finally, we gather some of the most recent advances from this topic.