Notes on the resolvent set

Hermitian Geometry on Resolvent Set

For a tuple A = (A1, A2, ..., An) of elements in a unital Banach algebra B, its projective joint spectrum P (A) is the collection of z ∈ C such that A(z) = z1A1 +z2A2 + · · ·+znAn is not invertible. It is known that the B-valued 1-form ωA(z) = A−1(z)dA(z) contains much topological information about the joint resolvent set P (A). This paper defines Hermitian metric on P (A) through the B-valued ...

Notes on Set Theory

We will now axiomatize set theory. From this point on, we cannot use any previous knowledge of mathematics, aside from providing intuition or motivation. Everyday words such as “finite” or “number” no longer have any technical meaning for us, except as specified by our axioms and definitions. These axioms and definitions will become our entire world, and we must reconstruct mathematics within i...

Notes on Constructive Set Theory

4 Operations on Sets and Classes 25 4.1 Class Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.2 Class Relations and Functions . . . . . . . . . . . . . . . . . . 26 4.3 Some Consequences of Union-Replacement . . . . . . . . . . . 27 4.4 Russell’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Subset Collection and Exponentiation . . . . . . . . . . . . . . 30...

Notes on Effective Descriptive Set Theory

Notes on point set topology, Fall 2010

The proofs that these maps are continuous are simple estimates that you probably remember from calculus. Since the continuity of all the maps we’ll look at in these notes is proved by expressing them in terms of the maps a and m, we include the proofs of continuity of a and m for completeness. Proof. To prove that the addition map a is continuous, suppose x = (x1, x2) ∈ R and > 0 are given. We ...