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
operato
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 th
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 str
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 )
=
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
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
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 th
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
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
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
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 ,
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
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 th
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 str
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 )
=
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
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
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 ,
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
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
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 st
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.
Howeve
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
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 th