Math 5340 Functional Analysis Fall 2006
Room/Time: Instructor: Email: Oce: Phone: Webpage: Oce Hours: Text: 3:004:00 PM MWF, Room Math 109 Dr. David S. Gilliam [email protected] Math Room 103 8067422580 ext. 265 http:/texas.math.ttu.edu/~gilliam/f06
BH 2. Every Hamel basis has the same number of elements. The number is called the dimension of
V , denoted by dim(V )3 .
BH 3. A normed space with a countable Hamel basis is not complete.4 Thus a Hamel basis in an
innite dimensional Banach space must be u
Thm 4.9 Given cfw_fj X linearly independent, there exists cfw_xj X such that fj (xi ) = ij .
Thm 4.10 If dim(X) < and M X, then M is closed and there exist a closed subspace N X
such that X = M N .
Thm 4.11 An linear operator K mapping X to Y is nite ra
Thm 2.10 X is a Banach space (even if X is not complete).
Thm 2.11
p
=
Thm 2.12 If f
q,
where 1/p + 1/q = 1.
p,
then there is a z
q
such that
f (x) =
xk zk ,
x
p
k=1
and
f = z q.
Thm 3.8 (Bounded Inverse Theorem) If X, Y are Banach spaces, and A B(X, Y
Moreover,
F (x)
x
xH
F = sup
x=0
Thm 2.2 Let N be a closed subspace of a Hilbert space H, and let x H\N . Set
d = inf x z .
zN
Then there exists a z N such that d = z x .
Thm 2.3 (Projection Theorem) Let N be a closed subspace of a Hilbert space H. Then
Theorems from Functional Analysis covered in Math 5340 Thm 1.1 Let X be a Banach space. If K is an operator mapping X to itself such that (a) K(u + v) = Ku + Kv (b) K(u) = K(u) (c) Ku M u
(d)
n=0
K nu <
for all u, v X. Then for each u X these is a unique
Molliers and Approximation by Smooth Functions with Compact Support
Let C (Rn ) be a nonnegative function with support in the unit ball in Rn . In particular we assume that (x) 0 for x Rn , (x) = 0 for x > 1, and
Rn
(x) dx = 1.
(1)
For example, we could
Theorems from Functional Analysis
covered in Math 5340
Thm 1.1 Let X be a Banach space. If K is an operator mapping X to itself such that
(a) K(u + v) = Ku + Kv
(b) K(u) = K(u)
(c) Ku M u
K nu <
(d)
n=0
for all u, v X. Then for each u X these is a unique
Semigroups of Operators in Banach Space
We begin with a definition Definition 1. A oneparameter family T (t) for 0 t < of bounded linear operators on a Banach space X is a C0 (or strongly continuous) Semigroup on X if 1. T (0) = I (the identity on X). 2.
Greens Function for Regular SturmLiouville Problems
We are interested in in solving problems like Ly := (py ) qy = f B1 y = 1 y(a) + 1 y (a) B2 y = 2 y(b) + 2 y (b). To this end we dene the operator Ly = (py ) qy under the assumption that = 0 is not an e
Examples: Inverses and Adjoints of a Simple Differential Operator
1. Let L = d/dx denote the differentiation operator acting in the Hilbert space H = L2 (a, b) where  < a < b < . 2. Tmax = L with D(Tmax ) = cfw_ AC(a, b) : H Important Note: Recall from r
Introduction to Sobolev Spaces on the Circle 1 Fourier Series
Recall that if L2 [0, 2] then has a Fourier expansion =
m=
am eimt ,
am =
1 2
2
(t)eimt ,
0
(1.1)
where the convergence of the infinite sum is in L2 [0, 2], i.e.,
N N
lim

m=N
am eimt = 0,
that
c1 x
1
x
0
c2 x
1
for all x X.
If X0 and X1 are equivalent then they are isomorphic via the identity map.
Furthermore, if there exists a constant C such that
x
0
C x
for all x X,
1
then X0 and X1 are equivalent.
BH 12. Let [cfw_xj , cfw_j ] be a ba