MATH 680A
December 6, 2009
Exam questions
1. Formulate and prove the contraction principle for maps of a metric space into itself.
Obtain the Picard-Lindelfs theorem as a corollary.
o
2. Formulate and prove the Arzela-Ascoli theorem. Describe the idea of
September 14, 2009
MATH 680A
Homework Assignment 1
Due Mon. Sep. 28, 2009, in class.
1. Given p, a prime number, dene the p-adic distance for x, y Q :
dp (x, y ) = pvp (xy) ,
x = y;
dp (x, y ) = 0,
x = y,
where vp (z ) = vp (r) vp (q ) for z = r/q , and v
October 14, 2009
MATH 680A
Homework Assignment 2
Due Wed. Oct. 28, 2009, in class.
1. If X is a normed space, the closure of any subspace is a subspace.
2. Let X be a Banach space.
(a) If T L (X, X ) and I T
invertible; in fact, the series
< 1 where I is
November 13, 2009
MATH 680A
Homework Assignment 3
Due Mon. Nov. 30, 2009, in class.
1. Show that every closed convex set K in a Hilbert space has a unique element of
minimum norm. [Adapt the proof of the Lemma on Orthogonal Projection which
addresses the
December 6, 2009
MATH 680A
Homework Assignment 4
1. Let A be a linear bounded operator on a Banach space. Show that if A is degenerate,
i. e. the range of A is nite-dimensional then A is compact.
2. Show that if An is a sequence of compact operators on a