User:TakuyaMurata/Functional Analysis

Intro
Functional Analysis can mean many things depending on which mathematician you ask. The core of the subject, however, is to study linear spaces with some topology which allows us to do analysis; ones like spaces of functions, spaces of operators acting on the space of functions, etc.

Hilbert space
A Banach space is called a Hilbert space if for each ordered pair $$(x, y)$$ of elements in the space there is a unique complex number called an inner product of $$x$$ and $$y$$, denoted by $$\langle x, y \rangle$$,