Outline of Material for Qualifying Test, Real Analysis,
2011
Sections of Rudin and the notes (with exceptions noted below).
(1)
(2)
(3)
(4)
(5)
Rudin chapters 1,2,3,4,6,8
Notes on molliers
notes of completions of metric spaces
course notes, chapters 1,2,4
Mollifiers and Smooth Functions
We say a function f from R C is C (or simply smooth ) if all its
derivatives to every order exist at every point of R. For f : Rk C,
we say f is C if all partial derivatives to every order exist and are
continuous.
Proposit
Some notes on completions
The completion of a metric space (X, d) is a complete metric space
(Y, D) together with an injective isometry X Y so that the closure
X = Y . Recall a map h : X Y is an isometry if for all x, y X ,
d(x, y ) = D(h(x), h(y ). We wi
A result for 8.6 in Rudin, Real Analysis II, Spring, 2011.
(Rudin 8.6) (Polar coordinates in Rk ). Let Sk1 the unit sphere in
R , i.e., the set of all u Rk whose distance from the origin 0 is 1.
Show that every x Rk , except for x = 0, has a unique repres