An Introduction to Analysis

Preface

I am putting some materials that seem to be beyond the scope of the book Real Analysis, in particular some functional and topological stuff that analysts use today (e.g., complex analysis, a partition of unity, functional analysis). No attention has been given with regard to the organization of the contents as I am not sure at this point. Also, no care and error check have been done; so do not trust the materials presented in the book blindly. -- Taku 10:46, 22 February 2006 (UTC)

Please note that the book is far from complete; some definitions are maybe missing, many proofs unfinished or erroneous, the structure of the book incoherent etc. -- Taku 07:14, 3 May 2007 (UTC)

0. Notation

The following notation will be in use.

• $E\subset \subset G$ : $E\subset K\subset G$ for some $K$ compact.
• $f\ll G$ : $\text{supp }f\subset G$.
• $𝒜(\mathrm{\Omega })$ = all functions analytic in $\mathrm{\Omega }$.
• $ℬ(\mathrm{\Omega })$ = all bounded linear operators in $\mathrm{\Omega }$.
• ${𝒞}^k(G)=$ all functions that are $k$-times continuously differentiable in $G$. We symbolically let $k=\mathrm{\infty }$ when the function is infinitely differentiable.
• $u^{(k)}={\left(\frac{\partial }{\partial z}\right)}^{k}u$.
• $\{fR0\}=\{z:f(z)R0\}$ where R may be =, <, etc.
• $\text{supp }f$ = the closure of $\{f=0\}$. (where the closure is taken is "usually" understood in context)
• $i{d}_E$ = the identity morphism (e.g., function, operator) on $E$.
• $bE=\overline{E}\int E$; i.e., the boundary of E (To avoid confusion, we reserve $\partial$ for differential stuff.)

We also assume:

• $\mathbb{N}$ does not include 0.
• $0\ne 1$. Such insanity does not exist in this book!
• In this book, an analytic function is always complex-valued.
• By paths, we always means a piecewise ${𝒞}^1$ curves. (See Chapter 2)

1. Sets and topology

• Sets and topology
  • Set theory
  • Topological space

2. Sequences of numbers

• /Sequences of numbers

3. Calculus

• Calculus

4. Differential forms

• /Differential forms