MATH 6220
HW6: CH3 BANACH SPACES
Due on Wednesday 29th April.
Exercise 1:
Let X be a vector space over R and V be an open convex subset containing 0.
Show that the function q(x) = infcfw_t 0 : x tV is a continuous semi norm on X.
A convex set V is balanc
NUTS AND BOLTS QUESTIONNAIRE
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MATH 6220 APPLIED FUNCTIONAL ANALYSIS
Instructor: Jeremie Brieussel (brieussel@math.cornell.edu, 255-4739)
Grader: Jason Anema (janema@math.cornell.edu)
Office hours announced soon.
Here is the program of the five first chapters of this course, which cove
MATH 6220
HW5: CH3 BANACH SPACES
Due on Monday 13th April.
Exercise 1:
We aim to prove the:
Theorem 0.1. Let (X, , ) be a measure space, then Lp (X, ) is a Banach space
for the norm:
Z
p1
p
|f |p =
|f | d .
X
We admit the fact that |.|p is a norm. We can
MATH 6220
CHAPTER 2 : OPERATORS ON HILBERT SPACES 3
Due on Monday 23rd March.
The number in italics refer to exercise and page in A Course in Functional Analysis J. B. Conway (2nd ed).
Exercise 1:(6p53)
This exercise shows that the assumption of injectivi
MATH 6220
CHAPTER 2 : OPERATORS ON HILBERT SPACES
Due on Monday 16th February.
The number in italics refer to exercise and page in A Course in Functional Analysis J. B. Conway (2nd ed).
Exercise 1:(9p23)
If H, K are two Hilbert spaces and U : H K is a sur
MATH 6220
FINAL EXAM
Due on Monday 11th May 2009.
Exercise 1:(20 points)
Let X be a normed vector space, X its dual.
We remind that a normed vector space X is separable if it admits a countable
dense family (xn )nN .
(1) Show that if X is separable, then
MATH 6220 APPLIED FUNCTIONAL ANALYSIS
SOLUTION OF PRELIMINARY EXAM
Exercise 1: (20 points)
Let T be a compact normal operator on a complex Hilbert space H. Show that
the two following properties are equivalent:
(1) For every in C, dim(Ker(T ) 1.
(2) There
MATH 6220
CHAPTER 1 : HILBERT SPACES
The number in italics refer to exercise and page in A Course in Functional Analysis J. B. Conway (2nd ed).
Exercise 1:(11p7)
Let H be a Hilbert space, (hn ) a sequence such that:
X
|hn | < ,
show that
P
n=0
n=0
hn conv
MATH 6220
CHAPTER 2 : OPERATORS ON HILBERT SPACES 2
Due on Monday 2nd March.
The number in italics refer to exercise and page in A Course in Functional Analysis J. B. Conway (2nd ed).
Exercise 1:(11p40)
Denote : l2 (Z) l2 (Z) the bilateral shift defined a
MATH 6220
PRELIMINARY EXAM 1
Due on Wednesday 25th March.
Remind N = cfw_0, 1, 2, 3, . . . .
Exercise 1:
Let T be a compact normal operator on a complex Hilbert space H. Show that
the two following properties are equivalent:
(1) For every in C, dim(Ker(T