Introduction to Analysis
Selected Homework Problems
1. Let S be a set and d a function from S×S intoR1 with the properties:
(i) d(x, y) = 0 if and only if x=y.
(ii) d(x, z)≤d(x, y) +d(z, y) for all x, y, z∈S.
Show that dis a metric and hence that (S, d) is a metric space.
2. Let A be the space of sequences x= {x1, x2,· · ·, xn,· · · } in which only a finite number of the xi are different from zero. In A, define d(x,y) by the formula
d(x,y) = max
1≤i<∞|xi−yi|.
(a) Show thatA is a metric space.
(b) Find a closed bounded set inA which is not compact.
3. Are closures and interiors of connected sets always connected? (Look at subsets ofR2.) 4. LetX be a metric space in which every infinite subset has a limit point. Prove thatX is
separable.
5. Let K be a compact subset of a metric space, and {Gα} be a covering of K with the property that every p∈K belongs to at least two sets in this covering. Show that {Gα} has a finite subcovering with the same property.
6. Show that a compact metric space X is connected if and only if it cannot be written as a union X =A∪B with infz∈A,b∈Bd(a, b)>0.
7. Prove that ifAis a connected set in a metric space andA ⊂B ⊂A, then¯ B is connected.
8. Suppose X is a complete metric space, and {Gn} is a sequence of dense open subsets of X. Prove Baire’s theorem, namely, that ∩∞1 Gn is not empty. (In fact, it is dense in X.) 9. Let f and g be continuous mappings of a metric space X into a metric space Y, and let
E be a dense subset of X. Prove thatf(E) is dense inf(X). If g(p) = f(p) for allp∈E, prove thatg(p) = f(p) for all p∈X.
10. Let X be a metric space, {sn} a sequence in X such thatlimn→∞d(sn, sn+1) = 0, and E the subsequential limit set of {sn}.
(a) Show that ifX =R, then E is connected.
(b) Show that ifX is compact, then E is connected.
11. A real-valued function f defined in (a, b) is said to be convex if f(λx+ (1−λ)y)≤λf(x) + (1−λ)f(y)
whenever a < x < b, a < y < b, 0 < λ < 1. Prove that every convex function is continuous. Prove that every increasing convex function of a convex function is convex.
(For example, if f is convex, so is ef.)
12. Suppose f is differentiable on [a, b], f(a) = 0, and there is a real number A such that
|f0(x)| ≤A|f(x)|on [a, b]. Prove that f(x) = 0 for all x∈[a, b]. Hint: Fix x0 ∈[a, b], let M0 = sup|f(x)|, M1 = sup|f0(x)|
for a≤x≤x0. For any such x,
|f(x)| ≤M1(x0−a)≤A(x0−a)M0. Hence M0 = 0 if A(x0−a)<1. That is , f = 0 on [a, x0]. Proceed.
13. (A mean-value theorem with possibly infinite end-points.)
Suppose −∞ ≤ a < b ≤ +∞, and f is differentiable function on (a, b) such that limx→af(x) = limx→bf(x). Show that there exists c∈(a, b) such that f0(c) = 0.
14. Let f :R→R be a differentiable function.
(a) Show that iff0 is bounded, then f is uniformly continuous.
(b) Show by example that the converse is not true.
(c) Show that if an example such as you are asked for in (b) is twice differentiable, then its second derivative must also be unbounded. Equivalently, (in view of (a)), show that a twice differentiable function R → R whose second derivative is bounded is uniformly continuous if and only if its first derivative is bound.
15. Consider the function f :R2 → R such that f,1 and f,2 exist and are bounded in a open ball Nr (0,0)
for somer >0.
(a) Show thef is continuous in Nr (0,0) .
(b) Decide whetherf is uniformly continuous in Nr (0,0) .
16. Let A be a closed region in Rk. Suppose that f : Rk → R is differentiable in a region containingA and that f has a maximum value at a point a ∈∂A. Show that dνf ≤0 at the point a where ν is the inward pointing unit normal to∂A at the pointa.
17. A C2 functionf :R2 →R is calledharmonic if D21f+D22f = 0.
Assume that (x0, y0) is a local maximum and f is harmonic. Prove that all second derivative of f vanish at (x0, y0).
18. Let E and F be two norm vector spaces with norms | · |E and | · |F respectively. We say that φ is a linear map from E toF if
φ(αx+βy) = αφ(x) +βφ(y), forx, y ∈E and α, β ∈R. Define
kφk:= inf{M
|φ(x)|F ≤M|x|E for all x∈E}.
Let L(E, F) be the space of all bounded linear maps from E toF. That is, L(E, F) ={φ
φ is a linear map fromE to F with kφ k<+∞}.
Prove that L(E, F) is a norm vector space with the norm k · k.
19. Let f : [0,∞) → R be a function which is continuous on [0,∞) and differentiable on (0,∞). Suppose that limx→∞f0(x) =Lis a finite number. Thenf is uniformly continuous on [0,∞).
20. Let X and Y be two metric spaces. Suppose that E is a disconnected compact set in X and f :X →Y is a continuous function with dY(f(x), f(y)) ≥ 12dX(x, y). Then f(E) is disconnected in Y.
