# Banach Lecture 5

Banach5

### Excerpts

Lemma I.7.1 $B$ is unital and $J$ is a proper ideal.  Then $\mbox{dist}(1,J) \ge 1$.

notes: the $r$ defined and used in the proof is not necessarily the spectral radius, but surely at least as big so $r$ can be used as in the proof.

Question: why does proof of corollary 2 show continuity?

$B$ is unital for the rest of the lecture.

Thm I.7.2 (Gelfand – Mazur) Upto iso, $\mathbb{C}$ is the only normed division algebra over $\mathbb{C}$.

notes: is $a -c$ is not invertible in the completed space, it is certainly not invertible in the smaller space.

Thm I.7.1 Let $a \in B$.  It is invertible iff its not contained in a maximal ideal.  Also $sp(a) = \{\phi(a) : \phi \in M(B)\}$ hence there exists multiplicative linear funtionals (since the spectrum is non empty).