Functional Analysis

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 convergence of ∑∞

n=1 ‖yn‖ may not imply the convergence of

∑∞ n=1 yn. (Hint: Look at the previous problem.)