# Banach Lecture 5

### Excerpts

Lemma I.7.1 is unital and is a proper ideal. Then .

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

Question: why does proof of corollary 2 show continuity?

is unital for the rest of the lecture.

Thm I.7.2 (Gelfand – Mazur) Upto iso, is the only normed division algebra over .

notes: is is not invertible in the completed space, it is certainly not invertible in the smaller space.

Thm I.7.1 Let . It is invertible iff its not contained in a maximal ideal. Also hence there exists multiplicative linear funtionals (since the spectrum is non empty).

## About this entry

You’re currently reading “Banach Lecture 5,” an entry on Math Meandering

- Published:
- September 4, 2009 / 1:01 pm

- Category:
- Banach Algebras (Course)

- Tags:

## No comments yet

Jump to comment form | comment rss [?] | trackback uri [?]