2.12. EXERCISES 75

2. Say ∥·∥ ,∥·∥1 are two equivalent norms. Explain carefully why if x is a limit point ofa set A with respect to ∥·∥ then it is also a limit point with respect to ∥·∥1. Also showthat if xn→ x with respect to ∥·∥ , then the same is true with respect to ∥·∥1.

3. If you have X a normed linear space and Y is a Banach space,(complete normed linearspace) show that L (X ,Y ) is a Banach space with respect to the operator norm.

4. If X ,Y are normed linear spaces, verify that A : X → Y is in L (X ,Y ) if and only ifA is continuous at each x ∈ X if and only if A is continuous at 0.

5. Generalize the root test, Theorem 1.12.1 to the situation where the ak are in a com-plete normed linear space.

6. Suppose X is a Banach space and {Bn} is a sequence of closed sets in X such thatBn ⊇ Bn+1 for all n and no Bn is empty. Also suppose that the diameter of Bn con-verges to 0. Recall the diameter is given by diam(B)≡ sup{∥x− y∥ : x,y ∈ B} . Thusthese sets Bn are nested and diam(Bn)→ 0. Verify that there is a unique point in theintersection of all these sets.

7. If X is a Banach space, and Y is the span of finitely many vectors in X , show that Yis closed.

8. If X is an infinite dimensional Banach space, show that there exists a sequence{xn}∞

n=1 such that ∥xn∥ ≤ 1 but for any m ̸= n,∥xn− xm∥ ≥ 1/4. Thus in infinitedimensional Banach spaces, closed and bounded sets are no longer compact as theyare in Fn.

9. In the proof of the fundamental theorem of algebra, explain why there exists z0 suchthat for p(z) a polynomial with complex coefficients, |p(z0)|= minz∈C |p(z)|> 0

10. Explain why a compact set in R has a largest point and a smallest point. Now iff : K→ R for K compact and f continuous, give another proof of the extreme valuetheorem from using that f (K) is compact.

11. Generalize Theorem 2.5.34 to the case where fn : S→ T where S,T are metric spaces.Give an appropriate definition for uniform convergence which will imply uniformconvergence transfers continuity from fn to the target function f .

12. A function f : X → R for X a normed linear space is lower semicontinuous if,whenever xn → x, f (x) ≤ liminfn→∞ f (xn) It is upper semicontinuous if, wheneverxn→ x, f (x)≥ limsupn→∞ f (xn) Explain why, if K is compact and f is upper semi-continuous then f achieves its maximum and if K is compact and f is lower semi-continuous, then f achieves its minimum on K.

13. Suppose fn : S→Y where S is a nonempty subset of X a normed linear space and sup-pose that Y is a Banach space (complete normed linear space). Generalize the theo-rem in the chapter to this case: Let fn : S→Y be bounded functions: supx∈S | fn (x)|=Cn < ∞. Then there exists bounded f : S→ Y such that limn→∞ ∥ f − fn∥ = 0 if andonly if { fn} is uniformly Cauchy. Also show that BC (S;Y ) is a Banach space.

14. Show that no interval [a,b] ⊆ R can be countable. Hint: First show [0,1] is notcountable. You might do this by noting that every point in this interval can be written