1/17/2012:

Analysis = Algebra + Infinity

Z, Q, R, C

Groups, Fields

Ex. Newton's method for y=x^2-2

Ex. Iterative solution of  y'=y, y(0)=1

These examples show the value of iterative solutions and also alert to the problem that the space in which we are seeking solutions "may have holes".

Ch. 1 The Real and Complex Number Systems                                      HW:   p21 / 1-5 try

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Order: trichotomy (x>y or x>y or x=y) and transitivity

ordered field - order and algebra mesh nicely

Ex. Q and R are ordered fields

Ex. Z_p and C are not 

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upper bounds, lower bounds, least upper bound (call it sup), greatest lower bound (call it inf)                            p21 / 6-9

Def. R is the (one and only) complete ordered field.

Not: existence and uniqueness are not obvious

A preview of things to come: Dirac delta function delta(x) - an "infinitely thin and infinitely high spike with area =1"              

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More on the delta(x) function:

intuitively,  int_R f(x) delta(x) dx = f(0) for all nice functions f.

What do we mean by nice functions f?

Counterexamples: functions which are discontinuous at 0, such as the unit step function U(x) and sin(1/x).

Why delta(x) is not a function: R -> R?

What is it then: a distribution; a functional (a rule that maps a function into a real number; it maps f into f(0) )

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Reading ahead is always an excellent idea ... casually browse through the book and get a feel for where this all leads to:
Ch. 9  Functions of Several Variables
Ch. 10 Integration of Differential Forms
Ch. 11 The Lebesgue Theory
Hilbert Space