I have continued my work on Belton http://www.maths.lancs.ac.uk/~belton/www/notes/fa_notes.pdf I finished off exercises 5.3-5.17. Primarily these were concerned with topological vector spaces (a Hausdorff topology on a vector space that makes the addition and scalar multiplication functions continuous), locally convex spaces, separating families of linear functionals ($M\subset X'$ is separating if for all $x\in X$, $\exists\,\phi\in M$ such that $\phi(x)\neq 0$). Also a number of results were derived that concerned the existence of functionals which were dominated on one set by another (e.g. 5.12). Finally some exercises on extreme points; for example every unit vector in a Hilbert space is an extreme point of the closed unit ball $B_1^H[0]$.

This section will be revised when I finish Belton. In particulat need to draw a scheme which relates the canonical topologies. Belton introduces them as initial topologies (generated by a family of functions) – the “old” terminology was the weak topology (generated by a family of functions). Also I will relook at the theorems to get a feeling for why and where particular conditions need to be satisfied (e.g. does the set need to be convex, compact, closed, connected?; does the space have to be locally convex, Hausdorff?, etc).

Having finished that section I began a study of normed algebras (vector spaces with an associative multiplication and submultiplicative norm). I saw that every finite dimensional algebra is isomorphic to a subalgebra of $M_{n}(\mathbb{F})$. I saw a number of examples of function spaces… basically it was “An Introduction to Normed Algebras” and it is fairly straightforward with some very nice results such as the Gelfand-Mazur Theorem. I have done exercises 6.1-6.6 of 8.

The final section of Belton is on Characters and Maximal Ideals. When I finish this I will be looking at http://books.google.com/books?id=emNvQgAACAAJ&dq=gerald+murphy+c*+algebras+and+operator+theory&hl=en&ei=6p_OTIfkJNXt4gaXv5n_Aw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA Alot of the stuff is in Belton so hopefully I can run through this text reasonably quickly.