Algebra 1

Course notes — Banach spaces, the big three theorems, and spectral theory.

Last updated 27 June 2026

Scope

Notes following the functional analysis lecture course. Core results covered:

Hahn–Banach: every bounded functional on a subspace extends to the whole space without increasing its norm.
Open mapping: a surjective bounded operator between Banach spaces is open.
Uniform boundedness: pointwise-bounded families of operators are norm-bounded.

A normed space $X$ is Banach if it is complete. The dual $X^* = B(X, \mathbb{R})$ is always Banach, even when $X$ is not, since

\|f\| = \sup_{\|x\| \le 1} |f(x)|

inherits completeness from $\mathbb{R}$ .

(Lecture-by-lecture notes to be added as separate pages in this section.)