Math 205b Homework 1 Solutions
January 18, 2013
Problem 1 (R-S, I.3) Let > 0. xn is a Cauchy sequence, so there is some k1 such that
for all m, n > k1 , (xm , xn ) < 2 . xn(i) x , so there is some k2 such that for all i k2 ,
(xn(i) , x ) < 2 .
Let k = max
Math 205b Homework 8 Solutions
March 12, 2013
Problem 1. (R-S, VI.10)
We rst claim that T is bounded. Indeed,
x
|f (t)|dt |f |C 0 x |f |C 0 ,
|T f (x)|
0
so T is bounded, and |T | 1. If f (x) = 1, then T f (x) = x, so that |T f | = 1, so |T | = 1.
We cla
MATH 205B: PROBLEM SET 7
DUE THURSDAY, FEBRUARY 28, 2013
Problem 1. (Reed-Simon V.9) Prove Theorem V.2.
Problem 2. (cf. Reed-Simon V.17) Let X, Y be locally convex spaces, and let X , Y be their
topological duals. Suppose that T : X Y is continuous and li
Math 205b Homework 7 Solutions
March 3, 2013
Problem 1 (R-S V.9)
We start by showing that T : X Y is continuous if and only if T is continuous at 0.
One direction is obvious, so we assume T is continuous at 0. Let x X , then we want to
show that T is cont
MATH 205B: PROBLEM SET 6
DUE THURSDAY, FEBRUARY 21, 2013
Problem 1. Suppose that (X, d) is a metric space.
(1) Suppose that f : [0, ) [0, ) satises f (0) = 0, f (x) > 0 if x > 0, f is increasing
(i.e. x y implies f (x) f (y ) and f is subadditive: f (x +
Math 205b Homework 6 Solution
February 19, 2013
Problem 1.
(1) f d 0, and if f (d(x, y ) = 0, then d(x, y ) = 0, so x = y , so f d is positive denite.
f d is clearly symmetric, and if x, y, z X , then d(x, z ) d(x, y ) + d(y, z ) and so
f (d(x, z ) f (d(x
MATH 205B: PROBLEM SET 5
DUE THURSDAY, FEBRUARY 14, 2013
Problem 1. (Reed-Simon IV.40) Let X be an innite dimensional Banach space
with the weak topology. Prove that the closure of the unit sphere is the unit ball.
Problem 2. Suppose X, Y are Banach space
Math 205b Homework 5 Solutions
February 11, 2013
Problem 1 (R-S IV.40)
Let B denote the unit ball cfw_x X : |x| 1 and S denote the unit sphere. S B , so
S B in X . Now let x B . We want to show that any neighborhood of x intersects S .
We have a base of t
MATH 205B: PROBLEM SET 4
DUE THURSDAY, FEBRUARY 7, 2013
Problem 1. Suppose that (X, d) is a metric space. A G set in X is a subset A of X such
that there exist open sets On , n = 1, 2, . . ., such that On = A.
n=1
(1) Suppose f : X R. Show that the set of
Math 205b Homework 4 Solutions
January 30, 2013
Problem 1
(1) First consider the set Aa = cfw_x : lim inf yx f (y ) > a. We claim that this set is open.
Indeed, if x Aa , then there is some > 0 such that if y B (x, ), f (y ) > a. So,
if we let y B (x, /2)
MATH 205B: PROBLEM SET 3
DUE THURSDAY, JANUARY 31, 2013
Problem 1. (Reed-Simon III.2)
(1) Prove that p (1 p < ) and c0 are separable, but
(2) Prove that s p for all p.
is not.
Problem 2. (Reed-Simon III.3) Prove that a normed linear space is complete if a
Math 205b Homework 3 Solutions
January 30, 2013
Problem 1 (R-S, III.2)
(1) Consider Vm , Vm = cfw_a = (a1 , a2 , . . .) : ak = 0 k > m. Note that Vm p for
all 1 p < , and Vm c0 . Note also that because all norms on Rm are equivalent,
we have that Vm is se
MATH 205B: PROBLEM SET 2
DUE THURSDAY, JANUARY 24, 2013
Problem 1. (cf. Reed-Simon, II.4) Suppose V is an inner product space either
over R or over C.
(1) Prove that the inner product can be recovered from the norm by the polarization identity:
1
(x, y )
Math 205b Homework 2 Solutions
January 24, 2013
Problem 1 (R-S, II.4)
(1) For the R case, we just expand the right hand side and use the symmetry of the inner
product:
1
|x + y |2 |x y |2 =
4
=
1
(x, x) + (y, y ) + (x, y ) + (y, x) (x, x) (y, y ) + (x, y
MATH 205B: PROBLEM SET 1
DUE THURSDAY, JANUARY 17, 2013
Problem 1. (Reed-Simon, Sec.I, no. 3) Suppose that xn is a Cauchy sequence in a
metric space (X, ). Suppose that for some subsequence, xn(i) , limi xn(i) = x .
Prove that limn xn = x .
Problem 2. (Re
MATH 205B: PROBLEM SET 8
DUE THURSDAY, MARCH 14, 2013
Problem 1. (Reed-Simon VI.10) Show that the spectral radius of the Volterra integral operator
x
(T f )(x) =
f (y ) dy
0
as a map on C ([0, 1]) is equal to 0. What is the norm of T ?
Problem 2. (Reed-Si