  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.

Functional Analysis