An Introduction to Analysis

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1 Preface
2 0. Notation
3 1. Sets and topology
4 2. Sequences of numbers
5 3. Calculus
6 4. Differential forms
7 6. Differential equations

[edit] 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)

[edit] 0. Notation

The following notation will be in use.

$E \subset \subset G$ : $E \subset K \subset G$ for some $K$ compact.
$f \ll G$ : $\mbox{supp }f \subset G$ .
$\mathcal{A}(\Omega)$ = all functions analytic in $Ω$ .
$\mathcal{B}(\Omega)$ = all bounded linear operators in $Ω$ .
$\mathcal{C}^k(G) =$ all functions that are $k$ -times continuously differentiable in $G$ . We symbolically let $k = \infty$ when the function is infinitely differentiable.
$u^{(k)} = \left({\partial \over \partial z}\right)^k u$ .
${f R 0} = {z : f (z) R 0}$ where R may be =, <, etc.
$supp f$ = 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 $E$ .
$bE = \bar E \ \int E$ ; i.e., the boundary of E (To avoid confusion, we reserve $\partial$ for differential stuff.)