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Appendix F
Multilinear algebra
Many of the algebraic constructions we shall carry out rely on fairly elementary
multilinear algebra. While the techniques are indeed fairly elementary, it is welladvised to be careful with which of
Chapter 3
Domains of holomorphy and notions of
convexity in Cn
In this chapter we study an important concept in holomorphic analysis, having to
do with the existence of extensions of holomorphic functions. The objects of interest
here are open subsets pos
Appendix A
General versions of the Chain Rule and the
Leibniz Rule
At various points in the text, we will wish to have on hand explicit formulae for
the Taylor series for compositions and products of smooth or holomorphic functions.
These are messy induct
Chapter 1
Holomorphic and real analytic calculus
In this chapter we develop basic analysis in the holomorphic and real analytic
settings. We do this, for the most part, simultaneously, and as a result some of the
ways we do things are a little unconventio
This version: 28/02/2014
Chapter 5
Holomorphic and real analytic jet bundles
In this chapter we study quite carefully the structure of jet bundles. Jets can be
thought of as a way of adapting the notion of Taylor series to the dierential
geometric setting
This version: 28/02/2014
Chapter 6
Stein and real analytic manifolds
In Corollary 4.2.11 we saw that there will generally be signicant restrictions on
the character of holomorphic functions on holomorphic manifolds. In Chapter 3 we
saw that there are doma
This version: 28/02/2014
Chapter 2
The Weierstrass Preparation Theorem and
applications
In this chapter we start by stating and proving the Weierstrass Preparation Theorem. This theorem has some deep consequences for the local structure of holomorphic
and
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Chapter 4
Holomorphic and real analytic differential
geometry
In this chapter we develop the basic theory of holomorphic and real analytic manifolds. We will be assuming that the reader has a solid background in basic smooth
diere
Appendix B
Convex analysis
In this appendix we review a few basic notions of convexity and related notions
that will be important for us at various times.
B.1 The Hausdorff distance
We begin with a fairly simple measure of closeness of sets.
B.1.1 Denitio