\chapter{Banach algebrae}

The operators from a normed linear space to itself form a Banach space themselves, but unlike other Banach spaces, composition of operators serves as a form of multiplication. Considering addition of operators, and the famous operator norm inequality, we can formalize the notion of a \emph{Banach algebra}; a Banach space that is an associative algebra where the "operator norm inequality" applies.


Though this formalism allows us to study this from merely an algebraic perspective, it is clearly made with the intent of studying operator spaces.

This type of algebraic structure is used for any Banach space for which a notion of multiplication of vectors exists

C-Algebra
C\*-Algebra

\section{Stone-Weierstrass theorem}

The familiar Weierstrass approximation theorem is generalized considerably by the \emph{Stone-Weierstrass theorem}; this theorem uses the theory of Banach algebrae to generalizee the function spaces used for approximating ('subalgebrae of the continuous functions that separate points' rather than space of polynomials) and applies the result for more general domain spaces (Hausdorff spaces rather than just $\mathbb{R}$).









