Math 571 - Functional Analysis I - Fall 2013
Homework 3
Assigned: Friday, September 13, 2013
Due: Friday, September 20, 2013
1. (# 4, Section 1.4) Show that a Cauchy sequence is bounded.
2. (# 5, Section 1.4) Is boundedness of a sequence in a metric space
Math 571 - Functional Analysis I - Fall 2013
Homework 7
Assigned: Saturday, October 26, 2013
Due: Friday, November 1, 2013
1. (# 2, Section 2.7) Let X and Y be normed spaces. Show that a linear operator
T : X Y is bounded if and only if T maps bounded set
Math 571 - Functional Analysis I - Fall 2013
Homework 12
Assigned: Monday, December 9, 2013
Due: Friday, December 13, 2013
1. (# 4, Section 3.6) Derive from the Parseval relation
| x, ek |2 = |x|2
k
the following formula (which is often called the Parseva
Math 571 - Functional Analysis I - Fall 2013
Homework 8
Assigned: Saturday, November 2, 2013
Due: Friday, November 8, 2013
1. (# 5, Section 2.8) Show that on any sequence space X we can dene a linear
functional f by setting f (x) = n (n xed), where x = (j
Math 571 - Functional Analysis I - Fall 2013
Homework 10
Assigned: Sunday, November 17, 2013
Due: Friday, November 22, 2013
1. (# 8, Section 3.2) Show that in an inner product space, x y if and only if
|x + y| |x| for all scalars .
2. (# 9, Section 3.2) L
Math 571 - Functional Analysis I - Fall 2013
Homework 9
Assigned: Saturday, November 9, 2013
Due: Friday, November 15, 2013
1. (# 9, Section 2.10) Show that a linear functional f on a vector space X is
uniquely determined by its values on a Hamel basis fo
Math 571 - Functional Analysis I - Fall 2013
Homework 6
Assigned: Saturday, October 5, 2013
Due: Friday, October 11, 2013
1. (# 1, Section 2.5) Show that Rn and Cn are not compact.
2. (# 3, Section 2.5) Give examples of compact and noncompact curves in th
Math 571 - Functional Analysis I - Fall 2013
Homework 1
Assigned: Friday, August 30, 2013
Due: Friday, September 6, 2013
Include a cover page and a problem sheet.
1. Show that the real line is a metric space.
2. Does d(x, y) = (x y)2 dene a metric on the
Math 571 - Functional Analysis I - Fall 2013
Homework 5
Assigned: Friday, September 27, 2013
Due: Friday, October 4, 2013
1. (# 2, Section 2.3) Show that c0 , the space of all sequences of scalars converging
to zero (see problem # 1, Section 2.3), is a cl
Math 571 - Functional Analysis I - Fall 2013
Homework 2
Assigned: Friday, September 6, 2013
Due: Friday, September 13, 2013
1. What is an open ball B(x0 ; 1) on R? In C? In C[a, b]? Explain Fig. 1.
Figure 1: Region containing the graphs of all x C[1, 1] w
Math 571 - Functional Analysis I - Fall 2013
Homework 4
Assigned: Friday, September 20, 2013
Due: Friday, September 27, 2013
1. (# 4, Section 2.1) Which is the following subsets of R3 constitute a subspace of
R3 ? [Here, x = (1 , 2 , 3 ).]
(a) All x with
Math 571 - Functional Analysis I - Fall 2013
Homework 11
Assigned: Monday, November 25, 2013
Due: Friday, December 6, 2013
1. (# 5, Section 3.4) If (ek ) is an orthonormal sequence in an inner product space
X, and x X, show that x y with y given by
n
y=
k