LECTURE #6.
Thursday, September 13, 2012.
COMPACTNESS
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
COMPACTNESS
DEF. A subset K of a MS X is called precompact iff every sequence cfw_xn
in K has a convergent subsequence cfw_xnk , i.e.,
(xnk , x) 0
LECTURE #1.
Tuesday, August 28, 2012
METRIC & NORMED SPACES
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
INTRODUCTION
In mathematical analysis we have dealt with different notions of limits
for sequences and functions. Sometimes we introduced dif
FALL 2010
MATH 6360
Yuliya Gorb
[email protected]
www.math.uh.edu/ gorb
Homework #2
due Thursday, September 30, 2010
Contraction Mapping Theorem. Normed Spaces. Linear Bounded Map.
1. Let L = cfw_x = (x1 , . . . , xn , . . .) 2 :
xn = 0.
n=1
(i) Is L a lin
FALL 2010
MATH 6360
Yuliya Gorb
[email protected]
www.math.uh.edu/ gorb
Homework #3
due Tuesday, October 25, 2010
Hilbert Spaces. Fourier Series. Linear Bounded Operators (cont).
1. Let A B (X ) and B B (X ), and also A1 B (X ) and B 1 B (X ). Show that th
FALL 2010
MATH 6360
Yuliya Gorb
[email protected]
www.math.uh.edu/ gorb
Homework #4
due Thursday, November 18, 2010
Special Linear Bounded Operators.
1. Prove or disprove the following statement. If xn x0 and yn y0 , then (xn , yn ) (x0 , y0 )
as n for xn
FALL 2010
MATH 6360
Yuliya Gorb
[email protected]
www.math.uh.edu/ gorb
Homework #5
due Thursday, December 02, 2010
Spectral Theory for Linear Bounded Operators.
1. If (A) then for all n one has n (An ).
2. If A B(H ) is self-adjoint then r(A) =: lim An 1/
Homework #4 ; M6361
Spring 2017.
Applicable Analysis; due 5.30pm, Thursday March 30th, 2017.
The notation here is that of the class - and the lecture notes. These problems may also
need standard definitions and results from differential multivariable calc
Homework #2 ; M6361
Spring 2017.
Applicable Analysis; due 5.30pm, Tuesday February 21st, 2017.
The notation here is that of the class - (and the lecture notes). From the lecture notes
do exercises 8.8, 9.2, 9.4 and 9.5. Also
Question 5. Suppose that f : (
Homework #6 ; M6361
Spring 2017.
Applicable Analysis; due 5.30pm, Thursday May 4th, 2017.
For this homework please do not discuss your work with anyone else. The notation is as
usual with e = (1, 1, . . . , 1) the n-vector of all 1s and |x| denotes the 2-
Homework #5 ; M6361
Spring 2017.
Applicable Analysis; due 5.30pm, Tuesday April 25th, 2017.
The notation here is that of the class - and the lecture notes.
Question 1. Show that the function f (x) := x ln x is continuous, convex and bounded
on (0,1). Find
'7-0 fnvbtys TVcfw_
/*
te parntti eh'L C
tr yil"A L Ex Acfw_."\
LC 2
ilrlcfw_u(.
$qE r
/,
E lALva/'e
p'*E*3b
j
a
J, J
t7
ed
xz+/(
Xt'
J:
7
^l
dv = els;r dx
"r 6P
t Jv= tts"'t(2t)gr
cosftcfw_
7
t+
(
cos'(t+)
.
+
i NT;
ul+
X=
A,vt,
,lx
?fnlg
u-5
X
FrA-rf
@q
LECTURE #5.
Tuesday, September 11, 2012.
COMPLETENESS
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
COMPLETNESS
DEF. A sequence cfw_xn X is a Cauchy sequence iff (xn , xm ) 0 as
n, m .
If cfw_xn is convergent then it is Cauchy.
If cfw_xn is Cau
LECTURE #4.
Thursday, September 06, 2012.
CONTINUITY, COMPLETENESS & NORMED SPACES
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
DEF. A mapping f of a metric space (X, X ) into a metric space (Y, Y )
is uniformly continuous iff
> 0 > 0 s.t. x, y
LECTURE #2.
Thursday, August 30, 2012
METRIC & NORMED SPACES (cont.)
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
Product spaces: Let (X, X ) and (Y, Y ) be two metric spaces. Then
dene Z = cfw_(x, y )| x X, y Y . Using X and Y we can dene
a metr
LECTURE #10.
Thursday, September 27, 2012.
HILBERT SPACES
c
Yuliya Gorb,
[email protected],
gorb
www.math.uh.edu/
INNER PRODUCT SPACES
DEF. A (linear) normed space X is called an inner product space over a
space K (where K cfw_R, C) iff there is a function
LECTURE #9.
Tuesday, September 25, 2012.
CONTRACTION MAPPING THEOREM
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
CONTRACTION MAPPING THEOREM
DEF. Let (X, ) be a metric space, and f : X X . The mapping f is a
contraction iff
k R : 0 k 1 s.t. (f
LECTURE #11.
Tuesday, October 02, 2012.
HILBERT SPACES
c
Yuliya Gorb,
[email protected],
gorb
www.math.uh.edu/
BEST APPROXIMATION THMs
DEF. Let
called a
(X, (, )
be an inner product space over
K cfw_R, C. X
is
Hilbert space, iff is is a complete normed spa
LECTURE #12.
Thursday, October 04, 2012.
ORTHONORMAL BASES
c
Yuliya Gorb,
[email protected],
gorb
www.math.uh.edu/
THM. Let (X, (, ) be an inner product space and M X be a nonempty subset. Let x X . If
1. M is complete, i.e. every Cauchy sequence cfw_xn M
LECTURE #13.
Tuesday, October 09, 2012.
ORTHONORMAL BASES
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
THM. Let cfw_e A be an orthonormal set of points in an inner product
space X . Then cfw_e are linearly independent.
DEF. System of elements cf
LECTURE #8.
Thursday, September 20, 2012.
NORMED SPACES
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
1-DIM SOBOLEV SPACES
Consider u L1 [a, b]
DEF. A function u L1 [a, b] is weakly differentiable iff there exists v
L1 [a, b] called a weak deriva
LECTURE #7.
Tuesday, September 18, 2012.
NORMED SPACES
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
NORMED LINEAR SPACES
DEF. A normed (linear) space (X, ) is a linear (vector) space X on
there is dened a function x x , called a norm, having prop
LECTURE #3.
Tuesday, September 04, 2012.
METRIC & NORMED SPACES (cont.)
c Yuliya Gorb,
[email protected],
www.math.uh.edu/gorb
DEF. A linear space X over the scalar eld R (or C) is a set of points (or
vectors) where operations of vector addition (x, y ) x
Homework #3 ; M6361
Spring 2017.
Applicable Analysis; due 5.30pm, Tuesday March 21st, 2017.
The notation here is that of the class - (and the lecture notes).
Question 1.
Let be the closed set whose boundary is a triangle in the plane whose
vertices are (0