FUNCTIONAL ANALYSIS LECTURE NOTES:
ADJOINTS IN BANACH SPACES
CHRISTOPHER HEIL
1. Adjoints in Banach Spaces
If H , K are Hilbert spaces and A B(H, K ), then we know that there exists an adjoint
operator A B(K, H ), which is uniquely dened by the condition
Homework 4
1. Let X and Y be normed spaces, T B (X, Y ) and cfw_xn a sequence in X . If xn x
weakly, show that T xn T x weakly.
Solution: We need to show that
g (Tn x) g (T x) g Y .
It suces to do this when g
Y
= 1. Observe that
|g (T xn ) g (T x)| = |g
Homework 5
1. Let H be the Hilbert space
2
. Let Sn :
2
2
be dened by:
Sn (x) = (0, . . . , 0, x1 , . . . , xn , . . .)
where we have shifted the sequence x to the right by n zeros. Show that Sn 0 strongly.
Solution: Note that
Sn x, y
2
xj y j + n .
=
j =
Homework 6
1. Let S, T B (X ) show that for any (T ) (S ) that
R (T ) R (S ) = R (T )(S T )R (S ).
Solution: Note that
T R (T ) = I
R (S )S = I.
Then we have
R (S )(T S )R (T ) = R (S )(T S )R (T )
= (R (S )T I )R (T )
= R (S ) R (T )
2. Let X be a comple
Homework 7
1. If X and Y are normed spaces, and T1 and T2 are compact linear operators, show that
T1 + T2 is a compact linear operator. Show that the compact linear operators from X into
Y constitute a subspace K (X, Y ) of B (X, Y ).
Solution: There are
Homework 8
1. Prove that if f L1 (Rn ) then f is uniformly continuous on Rn .
Solution: Note that we have
f (y + h) f (y ) =
f (x) e2ix(y+h) e2ixy dx
Rn
f (x)e2ixy (e2ixh 1)dx
=
Rn
Thus we have that
f (y + h) f (y )
|f (x)| e2ixh 1 dx
Rn
It suces to show
Math 6338 : Real Analysis II
Mid-term Exam 1
13 February 2012
Instructions: Answer all of the problems.
1. Let X be a real normed space. Suppose that C X is a convex subset which contains 0
and has the property that
tC = X
t>0
For every x X dene
QC (x) =
Math 6338 : Real Analysis II
Mid-term Exam 2
09 April 2012
Instructions: Answer all of the problems.
1. Suppose that L (R). Let M : L2 (R) L2 (R) be the operator:
M (f ) = f.
The essential range, R of a function L (R) is the set of all w C such that
|cfw_
Math 6338 : Real Analysis II
Final Exam
Due: Friday May 4 2012 at 5:00pm
Instructions: Answer all of the problems. You may use course notes, but other sources are
not permitted.
1. Let H be a complex separable Hilbert space. A linear operator T : H H is c
Homework 1
1. Let u be a semi-inner product on H and put N = cfw_x H : u(x, x) = 0. Show that
(a) Show that N is a linear subspace of H ;
(b) Show that if x + N, y + N u(x, y ) for all x + N and y + N in the quotient space
H/N , then , is a well-dened inn
Homework 2
1. Let X and Y be Hilbert spaces over C. Then a sesquilinear form h on X Y is a mapping
h : X Y C such that for all x1 , x2 , x X , y1 , y2 , y Y and all scalars , C we have
(a) h(x1 + x2 , y ) = h(x1 , y ) + h(x2 , y );
(b) h(x, y1 + y2 ) = h(
Homework 3
1. If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear
operator.
Solution: Assuming that the inverse of T were dened, then we will have to have that
D(T 1 ) = Ran(T ). Suppose that cfw_un D(T 1 ) is a sequen
Homework 4
1. Let X and Y be normed spaces, T B (X, Y ) and cfw_xn a sequence in X . If xn x
weakly, show that T xn T x weakly.
Solution: We need to show that
g (Tn x) g (T x) g Y .
It suces to do this when g
Y
= 1. Observe that
|g (T xn ) g (T x)| = |g
Homework 5
1. Let H be the Hilbert space
2
. Let Sn :
2
2
be dened by:
Sn (x) = (0, . . . , 0, x1 , . . . , xn , . . .)
where we have shifted the sequence x to the right by n zeros. Show that Sn 0 strongly.
Solution: Note that
Sn x, y
2
xj y j + n .
=
j =
Homework 6
1. Let S, T B (X ) show that for any (T ) (S ) that
R (T ) R (S ) = R (T )(S T )R (S ).
Solution: Note that
T R (T ) = I
R (S )S = I.
Then we have
R (S )(T S )R (T ) = R (S )(T S )R (T )
= (R (S )T I )R (T )
= R (S ) R (T )
2. Let X be a comple
Homework 7
1. If X and Y are normed spaces, and T1 and T2 are compact linear operators, show that
T1 + T2 is a compact linear operator. Show that the compact linear operators from X into
Y constitute a subspace K (X, Y ) of B (X, Y ).
Solution: There are
Homework 3
1. If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear
operator.
Solution: Assuming that the inverse of T were dened, then we will have to have that
D(T 1 ) = Ran(T ). Suppose that cfw_un D(T 1 ) is a sequen
Homework 2
1. Let X and Y be Hilbert spaces over C. Then a sesquilinear form h on X Y is a mapping
h : X Y C such that for all x1 , x2 , x X , y1 , y2 , y Y and all scalars , C we have
(a) h(x1 + x2 , y ) = h(x1 , y ) + h(x2 , y );
(b) h(x, y1 + y2 ) = h(
FUNCTIONAL ANALYSIS LECTURE NOTES:
ADJOINTS IN HILBERT SPACES
CHRISTOPHER HEIL
1. Adjoints in Hilbert Spaces
Recall that the dot product on Rn is given by x y = xT y , while the dot product on Cn is
x y = xT y .
Example 1.1. Let A be an m n real matrix. T
FUNCTIONAL ANALYSIS LECTURE NOTES:
REFLEXIVITY OF Lp
CHRISTOPHER HEIL
Notation: When talking about reexivity, it is convenient to take the action of a functional on a vector x to be linear in both and x. Usually, we use the ordinary notation
(x) to denote
FUNCTIONAL ANALYSIS LECTURE NOTES:
QUOTIENT SPACES
CHRISTOPHER HEIL
1. Cosets and the Quotient Space
Any vector space is an abelian group under the operation of vector addition. So, if you are
have studied the basic notions of abstract algebra, the concep
1.3 Convolution
15
1.3 Convolution
Since L1 (R) is a Banach space, we know that it has many useful properties. In
particular the operations of addition and scalar multiplication are continuous.
However, there are many other operations on L1 (R) that we co
FUNCTIONAL ANALYSIS LECTURE NOTES:
WEAK AND WEAK* CONVERGENCE
CHRISTOPHER HEIL
1. Weak and Weak* Convergence of Vectors
Denition 1.1. Let X be a normed linear space, and let xn , x X .
a. We say that xn converges, converges strongly, or converges in norm
Math 6338 : Real Analysis II
Mid-term Exam 1
13 February 2012
Instructions: Answer all of the problems.
1. Let X be a real normed space. Suppose that C X is a convex subset which contains 0
and has the property that
tC = X
t>0
For every x X dene
QC (x) =