Math 511 Problem Set 4, due September 21 Note: Problems 1 through 7 are the ones to be turned in. The remainder of the problems are for extra functional analytic goodness. 1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a linearly independent subset of C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly independent set in C[a,b]. Give an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . . .}. 2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 . 3. Consider C[0, 2] with the function ‖ ·‖1 defined by ‖f‖1 = ∫ 2 0 |f(x)|dx, for f ∈ C[0, 2]. (a) Prove that ‖ ·‖1 is a norm. (b) Prove that the normed linear space (C[0, 2],‖·‖1) is not complete (and thus not a Banach space) by considering the sequence of functions fn(x) =   1, x ≤ 1 − 1 n n−nx, 1 − 1 n < x < 1 + 1 n −1, x ≥ 1 + 1 n . Show these are continuous functions, this sequence is a Cauchy sequence in the metric derived from ‖ ·‖1, but that this sequence does not converge in C[0, 2] with this metric. 4. Let V be a vector space over R or C. A subset A ⊆ V is convex if for any v,w ∈ A and any λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting v and w is also in A. (a) Let W be a vector subspace of V . Show that W is convex. (b) Let X be a normed linear space. Show that the unit ball B1(0) is convex. 5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the definition of c) and so is c0, the set of all sequences (xn) so that limn→∞ xn = 0. 6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth place and 0 in all othe coordinates. Show that {en : n ∈ N} is a Schauder basis for lp. 7. Now if X is a Banach space and (yn) a sequence in X, prove that ∑∞ n=1 ‖yn‖ < ∞ does imply the convergence of ∑∞ n=1 yn. Thus in Banach spaces, absolute convergence implies convergence of the series. The following questions are for you to think about and not to be turned in. 1001. What is the completion of (0, 1) as a metric subspace of R with the euclidean metric? Explain. 1002. Show that the discrete metric on a nontrivial vector space cannot be obtained from a norm. 1003. Show that if a normed vector space has a Schauder basis, then the space is separable. (You can use a similar argument to your proof that lp is separable for 1 ≤ p < ∞.) 1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p < ∞, and assume that 1 p + 1 q = 1 r . Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we define the componentwise product xy = (x1y1,x2y2, . . .), then ‖xy‖r ≤‖x‖p‖y‖q. You may assume that x ∈ lp and y ∈ lq, although this is not necessary. (Hint: 1 = 1p r + 1q r , and use the regular Hölder inequality on particular sequences). (Note: We can extend this to let p = r, and in this case q = ∞. The result will still hold.) 1005. Give an example of a subspace of l∞ which is not closed. Repeat for l2. (Hint: Look at problem 3, p. 70) 1006. Let X be a normed vector space. Show that the convergenc ...

1. Math 511 Problem Set 4, due September 21
Note: Problems 1 through 7 are the ones to be turned in. The
remainder of the problems are
for extra functional analytic goodness.
1. Fix a,b ∈ R with a < b. Show that {1, t, t2, . . . , tn} is a
linearly independent subset of
C[a,b]. From this conclude that {1, t, t2, t3, . . .} is a linearly
independent set in C[a,b]. Give
an example of a function f ∈ C[a,b] so that f /∈ span{1, t, t2, . .
.}.
2. Prove that if 1 ≤ p1 ≤ p2 ≤∞ then lp1 ⊆ lp2 .
3. Consider C[0, 2] with the function ‖ ·‖1 defined by
‖f‖1 =
∫ 2
0
|f(x)|dx, for f ∈ C[0, 2].
(a) Prove that ‖ ·‖1 is a norm.
(b) Prove that the normed linear space (C[0, 2],‖·‖1) is not
complete (and thus not a Banach
space) by considering the sequence of functions
fn(x) =

2. 1, x ≤ 1 − 1
n
n−nx, 1 − 1
n
< x < 1 + 1
n
−1, x ≥ 1 + 1
n
.
Show these are continuous functions, this sequence is a Cauchy
sequence in the metric
derived from ‖ ·‖1, but that this sequence does not converge in
C[0, 2] with this metric.
4. Let V be a vector space over R or C. A subset A ⊆ V is
convex if for any v,w ∈ A and any
λ ∈ [0, 1] then λv + (1 −λ)w ∈ A, i.e. the segement connecting
v and w is also in A.
(a) Let W be a vector subspace of V . Show that W is convex.
(b) Let X be a normed linear space. Show that the unit ball
B1(0) is convex.
5. show that c ⊆ l∞ is a vector subspace of l∞ (see 1.5-3 for the
definition of c) and so is c0, the
set of all sequences (xn) so that limn→∞ xn = 0.
6. Let 1 ≤ p < ∞ and en ∈ lp be the sequence with 1 in the nth
place and 0 in all othe coordinates.

3. Show that {en : n ∈ N} is a Schauder basis for lp.
7. Now if X is a Banach space and (yn) a sequence in X, prove
that
∑∞
n=1 ‖yn‖ < ∞ does imply
the convergence of
∑∞
n=1 yn. Thus in Banach spaces, absolute convergence implies
convergence
of the series.
The following questions are for you to think about and not to be
turned in.
1001. What is the completion of (0, 1) as a metric subspace of R
with the euclidean metric?
Explain.
1002. Show that the discrete metric on a nontrivial vector space
cannot be obtained from a norm.
1003. Show that if a normed vector space has a Schauder basis,
then the space is separable. (You
can use a similar argument to your proof that lp is separable for
1 ≤ p < ∞.)
1004. Prove the general Hölder inequality: Suppose 1 ≤ r < p <
∞, and assume that
1

4. p
+
1
q
=
1
r
.
Show that for x = (x1,x2, . . .) and y = (y1,y2, . . .), and if we
define the componentwise product
xy = (x1y1,x2y2, . . .), then
‖xy‖r ≤‖x‖p‖y‖q.
You may assume that x ∈ lp and y ∈ lq, although this is not
necessary. (Hint: 1 = 1p
r
+ 1q
r
, and
use the regular Hölder inequality on particular sequences).
(Note: We can extend this to let p = r, and in this case q = ∞.
The result will still hold.)
1005. Give an example of a subspace of l∞ which is not closed.
Repeat for l2. (Hint: Look at
problem 3, p. 70)

5. 1006. Let X be a normed vector space. Show that the
convergence of
∑∞
n=1 ‖yn‖ may not imply
the convergence of
∑∞
n=1 yn. (Hint: Look at the previous problem.)
Math 511 Problem Set 5, due October 5
1. Let X be a normed linear space and Y a finite dimensional
subspace of X so that Y 6= X.
Prove that there is a z ∈ XY with ‖z‖ = 1 such that for any y ∈
Y , ‖z − y‖ ≥ 1. (Hint:
Pick some z0 ∈ XY , and let Z = span{Y, z0}. Since Y is finite
dimensional, then Z is finite
dimensional, and use Riesz’s Lemma and the compactness of the
closed unit ball of Z.)
2. Suppose that X and Y are vector spaces and T : X → Y is an
invertible linear operator.
Show that if {x1, x2, . . . , xn} is a linearly independent set in
X, then {Tx1, Tx2, . . . , Txn} is
linearly independent in Y .
3. Define the linear map T : l∞ → l∞ by Tx =
(
xk
k
)

6. , where x = (xk). Show that T is a bounded
linear operator and is injective. Is T−1 bounded? (Note: The
domain of T−1 may not be all
of l∞)
4. Let X, Y , and Z be normed linear spaces. For S ∈ B(X, Y )
and T ∈ B(Y, Z) define
the linear operator TS : X → Z by function composition: TSx =
T ◦ S(x). Prove that
‖TS‖≤‖T‖‖S‖.
5. Let X = R2 with the norm ‖ · ‖1 so that ‖(y, z)‖1 = |y| + |z|.
Fix some a ∈ R, and define
T ∈ B(X, X) by
T (y, z) =
[
1 a
0 1
] [
y
z
]
.
Prove that ‖T‖ = 1 + |a|. Observe the difference in this case with
the example from class in
that the norm of an operator depends on the norms of its domain
and codomain.
6. Consider the operator A : l2 → l2 given by x = (xn) ∈ l2
implies Ax = ( xnn ). Show that

7. range(A) is a dense subspace of l2 but not closed.
Math 511 Problem Set 6, due October 12
1. We say that a linear operator T : X → Y between two normed
spaces is bounded below if
there is some b > 0 so that for every x ∈ X, ‖Tx‖ ≥ b‖x‖.
Suppose that T ∈ B(X,Y ) is
bounded below by b > 0.
(a) Show that T is injective.
(b) Show that if X is a Banach space, then range(T) is a closed
subspace of Y .
(c) Show that T−1 : range(T) → X is a bounded linear operator
and ‖T−1‖≤ 1
b
.
2. Let X be a vector space over field F and Y a subspace. If x ∈
X, the coset of Y containing
x is the set x + Y = {x + y : y ∈ Y}. Denote the set of all cosets
of Y to be X/Y . Define
an addition and scalar multiplication on X/Y by if x1 + Y,x2 +
Y ∈ X/Y and k ∈ F, then
(x1 + Y ) + (x2 + Y ) = (x1 + x2) + Y and k(x1 + Y ) = (kx1) +
Y.
(a) Show that these operations are well-defined, and these make
X/Y into a vector space.

8. (b) Define the codimension of Y to be codim(Y ) = dim(X/Y ).
Prove that if f is a linear
functional on X, then codim(ker(f)) = 1.
3. Suppose that X is a linear space and Y is a subspace. You
showed above that X/Y is a
vector space. Now suppose additionally that X has a norm and Y
is a closed subspace.
Show that ‖x + Y‖ = inf{‖x + y‖ : y ∈ Y} defines a norm on X/Y
.
Note: It also follows that if X is a Banach space then X/Y is
also a Banach space. The
proof can be found online.
4. Let X = C[0, 1] with the supremum norm, ‖ · ‖∞. Consider the
functional φ defined for
f ∈ X by
φ(f) =
∫ 1
2
0
f(x) dx−
∫ 1
1
2
f(x) dx.
(a) Prove that φ is a bounded linear functional with norm 1.
(b) Show that if f ∈ X with ‖f‖ = 1 then |φ(f)| < 1.

9. (c) Briefly describe how this relates to Riesz’s lemma (2.5-4).
(Hint: φ is continuous, so
ker(φ) is a closed subspace of X.)
Math 511 Problem Set 7, due October 19 (can turn in through
Oct. 22)
A couple thoughts about proving things in inner product spaces:
• The norm and inner product are intimately linked. Don’t be
afraid to substitute one for the
other as needed.
• Often times it’s much easier to work with the square of the
norm rather than the norm itself
for calculations (so you can bring in the inner product).
Now the exercises:
1. If X is a real inner product space show that ‖x‖ = ‖y‖ implies
that 〈x+y,x−y〉 = 0. What
does this mean geometrically in X = R2? What does this
condition imply if X is complex?
2. Let X be an inner product space, y ∈ X, and (xn) a sequence
in X so that xn is orthogonal
to y for every n. Show that if xn → x, then x is orthogonal to y
as well.
3. Show that for a sequence (xn) in an inner product space, the
conditions ‖xn‖ → ‖x‖ and
〈xn,x〉→ 〈x,x〉 imply the convergence xn → x.

10. 4. Pythagorean theorem: Let X be an inner product space, and
x,y ∈ X. Prove that if
〈x,y〉 = 0, then ‖x‖2 + ‖y‖2 = ‖x + y‖2.
5. Show that in an inner product space that x is orthogonal to y
if and only if ‖x + αy‖≥‖x‖
for all scalars α.
(Hint: If 〈x,y〉 6= 0, then there is some scalar α with |α| = 1
and 〈x,αy〉 < 0. Then consider
the values of ‖x + αty‖, where t is a real parameter (double hint:
quadratic formula?))
6. (a) Suppose that X is a real inner product space. Prove the
polarization identity: for any
x,y ∈ X,
〈x,y〉 =
1
4
(
‖x + y‖2 −‖x−y‖2
)
(b) Suppose X is a real normed linear space such that the norm
satisfies the parallelogram
equality (page 130 in Kreyszig). Prove that if we define a
function 〈x,y〉 as in the
polarization identity above, this function is an inner product on
X so that for any
x ∈ X we have 〈x,x〉 = ‖x‖2. (This shows that an inner product
can be recovered from

11. its norm.)
(c) Something to think about but not turned in: If X is a
complex inner product
space, the corresponding polarization identity is: for any x,y ∈
X,
〈x,y〉 =
1
4
(
‖x + y‖2 −‖x−y‖2 + i‖x + iy‖2 − i‖x− iy‖2
)
.
And if a complex normed linear space obeys the parallelogram
equality, its inner product
can be recovered in a similar manner from the norm.