Math 497B Elements of Functional Analysis
Homework 6
Name: Zhaoning Yang
October 23, 2013
Problem 1 Let F be a monotonically non-decreasing function on [a, b]. Show that left and right limits
dened by
F (s 0) := lim F (t) and F (s + 0) := lim F (t)
ts
ts+
Math 497B Elements of Functional Analysis
Homework 3
Name: Zhaoning Yang
September 18, 2013
Problem 1
Let A be a linear operator on Rn given by the matrix (aik ).
(1) Find A if Rn is equipped with x
(2) Show that if Rn is equipped with x
Frobenious norm)
MATH 497B ELEMENTS OF FUNCTIONAL ANALYSIS
FINAL PREPARATION
ZHAONING YANG
1. Exam Questions
Soluion: Let (X, d) be a metric space, K X is said to be compact if every sequence in K has a convergent
subsequence. Another equivalent denition using open sets i
Final Exam, Solutions
Math 502, SPRING 2014
Solve three problems:
Problem 1. Show that if f is holomorphic in the open unit disk U and
there is a point z such that
f (z) + f (z) + f (z) + . . . +
converges then f (z) could be extended to a holomorphic fun
Math 497B Elements of Functional Analysis
Homework 4
Name: Zhaoning Yang
September 25, 2013
Problem 1
Show that (Rn , x
and (Rn , x 1 ) are dual to each other. Are they isometric?
)
Solution:
Proof. Let V = (Rn , x ) and W = (Rn , x 1 ), we want to show V
Math 497B Elements of Functional Analysis
Homework 5
Name: Zhaoning Yang
October 2, 2013
Problem 1
Show that C([0, 1]) equipped with the norm x
1
=
1
0 |x(t)|dt
is not complete.
Solution:
Proof. Dene for each n Z, xn : [0, 1] R by
xn (t) =
nt + 1
0
1
if 0
Math 497B Elements of Functional Analysis
Homework 9
Name: Zhaoning Yang
Due: Wednesday, November 11, 2013
Problem 1.
Consider the space Lp = Lp (T, ), where T = (0, ) R and is the Lebesgue measure
(a) Give examples of continuous functions which show that
Math 497B Elements of Functional Analysis
Homework 8
Name: Zhaoning Yang
November 7, 2013
Problem 1 Let (X, M, ) be a measure space. Suppose E M and f : E R is a measurable
function. Prove if f is nonnegative and E f d = 0 then f 0 almost everywhere on E.
Math 497B Elements of Functional Analysis
Homework 7
Name: Zhaoning Yang
October 31, 2013
Problem 1
Let (X, , M) be a measure space, show that
(1) If B A then (B) (A). If also (A) < then (A B) = (A) (B).
(2) If A1 A2 . . . , and A = An , then (A) = limn (
Math 497B Elements of Functional Analysis
Homework 2
Zhaoning Yang
September 10, 2013
Problem 1 Let (X, X ) and (Y, Y ) be metric spaces, and suppose f : X Y is a function. We dene
the graph of f to be the following
f = cfw_(x, y) X Y | y = f (x)
(1) Prov