MAST30026 Problem Sets

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 18 March 2014

Problem Set 1

  1. Check if the following functions are metrics on X.
    1. d(x,y)= |x2-y2| for x,yX=
    2. d(x,y)= |x2-y2| for x,yX=(-,0]
    3. d(x,y)= |arctanx-arctany| for x,yX=
  2. (French railroad metric) Let X=2 and let d be the usual metric. Denote by 0=(0,0) and define d0(x,y)= { 0 ifx=y; d(x,0)+ d(0,y) ifxy. Verify that d0 is a metric on X. (Paris is at the origin 0.)
  3. Let X=2. For x=(x1,x2) and y=(y1,y2) define d(x,y)= { 1/2 ifx1=y1 andx2y2 or ifx1y1 andx2=y2; 1 ifx1y1 andx2y2; 0 otherwise. Verify that d is a metric and that two congruent rectangles, one with base parallel to the x-axis and the other at 45 to the x-axis, have different “area” if d is used to measure the length of sides.
  4. Let (X,d) be a metric space. Consider the function f:[0,)[0,) having the following properties:
    1. f is non-decreasing, i.e. f(a)f(b) if 0a<b;
    2. f(x)=0 if and only if x=0;
    3. f(a+b)f(a)+f(b), a,b[0,).
    If x,yX define df(x,y)=f(d(x,y)). Show that df is a metric and that the functions f(t)=kt where k>0, f(t)=tα where 0<α1, and f(t)=t1+t for t0 have properties (a)–(c).
  5. (p-adic metric) Let p be a prime number. Define the p-adic absolute value function |·|p on by setting |x|p=0 when x=0 and |x|p=p-k when x=pk·mn where m,n are nonzero integers which are not divisible by p. Show that for x,y, |x+y|p max{|x|p,|y|p} and that d(x,y)=|x-y|p defines a metric on . In fact, d(x,z)max{d(x,y),d(y,z)}. If d satisfies this condition which is stronger than the triangle inequality then d is called an ultrametric.
  6. Let (Xi,di) be a metric space for 1in and let X=i=1nXi. Define d(x,y) = [i=1ndi(xi,yi)2]1/2, d(x,y) = max { di(xi,yi) |1in } , where x=(x1,,xn) and y=(y1,,yn)X. Verify that d and d are metrics on X.
  7. Fix a positive integer n. Denote by 𝒫n the real vector space of all polynomials p(x)=anxn+an-1xn-1++a1x+a0 with real coefficients ai. For p(x)=anxn+an-1xn-1++a1x+a0𝒫n set p=max { |a0|, |a1|,, |an| } Verify that · is a norm on 𝒫n.
  8. Let (Xn,dn), n, be a sequence of metric spaces and let X=nXn be the cartesian product of the Xn's. (The elements of X are of the form x=(x1,x2,) with xnXn.) For x,yX, define d(x,y)= n=1 12n dn(xn,yn) 1+dn(xn,yn) . Show that (X,d) is a metric space.
  9. Sketch the open ball B(0,1) in the metric space (3,di), where di is defined by d1(x,y) = |x1-y1|+ |x2-y2|+ |x3-y3| d2(x,y) = (x1-y1)2+ (x2-y2)2+ (x3-y3)2 d(x,y) = max { |x1-y1|, |x2-y2|, |x3-y3| } for x=(x1,x2,x3) and y=(y1,y2,y3)3.
  10. Set d(n,m)= |1n-1m| for n,m. Then d is a metric.
    1. Let P be the set of positive even numbers. Find diam(P) and diam(\P) in (,d).
    2. For a fixed n find all elements of B(2n,12n) and B(n,12n).

Problem Set 2

  1. Let X=2. For x=(x1,x2) and y=(y1,y2)X define dM(x,y)= { |x2-y2| ifx1=y1; |x1-y1|+ |x2|+ |y2| ifx1y1. Also define dK(x,y)= { x-y ifx=ty for somet; x+y otherwise where x=[i=1nxi2]1/2.
    (Can you give reasonable interpretations of the metrics dM and dK?)
    Study the convergence of the sequence xn in the spaces (X,dM) and (X,dK) if
    1. xn=(1n,nn+1);
    2. xn=(nn+1,nn+1);
    3. xn=(1n,n+1-n).
  2. Let {xn} and {yn} be sequences in a metric space (X,d) such that xnn and yny as n. Prove that d(xn,yn)d(x,y) as n.
  3. Metrics d and d defined on X are called Lipschitz equivalent if there exist positive constants m,M such that m·d(x,y) d(x,y) M·d(x,y) for all x,yX.
    1. Show that if d and d are Lipschitz equivalent, then they are equivalent. Give an example of X and two equivalent metrics on X which are not Lipschitz equivalent.
    2. For p1 and x,yn, the lp metric is defined by dp(x,y)= [i=1n|xi-yi|p]1/p =x-yp. Show that if p,q1, then dp and dq are Lipschitz equivalent. (Hint: compare these with d(x,y)=max{|x1-y1|,,|xn-yn|}.)
  4. Consider the set X=[-1,1] as a metric subspace of with the standard metric. Let
    1. A={xX|1/2<|x|<2};
    2. B={xX|1/2<|x|2};
    3. C={x|1/2|x|<1};
    4. D={x|1/2|x|1};
    5. E={x|0<|x|1and1/x}.
    Classify the sets in (a)–(e) as open/closed in X and .
  5. Consider 2 with the standard metric. Let
    1. A={(x,y)|-1<x1and-1<y<1};
    2. B={(x,y)|xy=0};
    3. C={(x,y)|x,y};
    4. D={(x,y)|-1<x<1andy=0};
    5. E=n=1{(x,y)|x=1/nand|y|n}.
    Sketch (if possible) and classify the sets in (a)–(e) as open/closed/neither in 2.
  6. Find the interior, the closure and the boundary of each of the following subsets of 2 with the standard metric:
    1. A={(x,y)|x>0andy0};
    2. B={(x,y)|x,y};
    3. C=AB;
    4. D={(x,y)|xis rational};
    5. E={(x,y)|x0andy1/x}.
  7. Let A be a subset of a metric space X. Is the interior of A equal to the interior of the closure of A? Is the closure of the interior of A equal to the closure of A itself?
  8. Consider a collection {Ai}iI of subsets of a metric space X. Show that iIAi (iIAi) iIAi iIAi (iIAi) iIAi iIAi iIAi
  9. Let (X,d) be a metric space. Show that if AX, then
    1. A=AA.
    2. A=A\A and A=A\A.
    3. A is closed if and only if A=A\A.
    4. A is open if and only if A=A\A.
  10. Let X and Y be metric spaces and A,B non-empty subsets of X and Y, respectively. Prove that
    1. If A×B is an open subset of X×Y, then A and B are open in X and Y, respectively.
    2. If A×B is an closed subset of X×Y, then A and B are closed in X and Y, respectively.

Problem Set 3

  1. Let d and d be equivalent metrics on X. Show that
    1. AX is closed in (X,d) if and only if A is closed in (X,d);
    2. AX is open in (X,d) if and only if A is open in (X,d).
  2. Show that if AX then diamA=diamA. Does diamA=diamA0?
  3. Let (X,dX) and (Y,dY) be metric spaces and A,B are dense subsets of X and Y, respectively. Show that A×B is dense in X×Y.
  4. Let C be the circle in 2 with the centre at (0,1/2) and radius 1/2. Let X=C\{(0,1)}. Define the function f:X by defining f(t) to be the point at which the line segment from (t,0) to (0,1) intersects X.
    1. Show that f:X and f-1:X are continuous.
    2. Define for s,t ρ(s,t)= f(s)-f(t) where · is the standard norm in 2. Show that ρ defines a metric on which is equivalent to the standard metric on .
  5. Let X=C[0,1]. Let F:X be defined by F(f)=f(0). Moreover, let d(f,g)=sup{|f(x)-g(x)|:x[0,1]} and d1(f,g)=01|f(x)-g(x)|dx.
    Is F continuous when X is equipped with (a) the metric d, (b) the metric d1?
  6. Let (X,dX) and (Y,dY) be metric spaces. Show that f:XY is continuous if and only if
    1. f(A)f(A) for all subsets A of X, or
    2. f-1(B)f-1(B) for all subsets B of Y.
  7. Let (X,d) be a metric space and let a be a fixed point of X. Show that |d(x,a)-d(y,a)| d(x,y) for all x,yX. Conclude that the function f:X defined by f(x)=d(x,a) is uniformly continuous.
  8. Which of the following functions are uniformly continuous?
    1. f(x)=sinx on [0,)
    2. g(x)=11-x on (0,1)
    3. h(x)=x on [0,)
    4. k(x)=sin(1/x), on (0,1)
  9. Which of the following sequences of functions converge uniformly on the interval [0,1]?
    1. fn(x)=nx2(1-x)n
    2. fn(x)=n2x(1-x2)n
    3. fn(x)=n2x3e-nx2
  10. Suppose that A is a dense subset of a metric space (X,d) and f:A is uniformly continuous. Show that there exists exactly one continuous function g:X satisfying g(x)=f(x) for xA.
    (Hint: You may need to use the completeness of .)

Problem set 4

  1. Suppose that {xn} and {yn} are Cauchy sequences in a metric space (X,d). Prove that the sequence of real numbers {d(xn,yn)} converges.
  2. Suppose that {xn} is a sequence in a metric space (X,d) such that d(xn,xn+1)2-n for all n. Prove that {xn} is a Cauchy sequence.
  3. Decide if the following metric spaces are complete:
    1. ((0,),d), where d(x,y)=|x2-y2| for x,y(0,).
    2. ((-π/2,π/2),d), where d(x,y)=|tanx-tany| for x,y(-π/2,π/2).
  4. Let X=(0,1] be equipped with the usual metric d(x,y)=|x-y|. Show that (X,d) is not complete. Let d(x,y)=|1x-1y| for x,yX. Show that d is a metric on X that is equivalent to d, and that (X,d) is complete.
  5. Suppose that (X,d) and (Y,d) are metric spaces and that f:XY is a bijection such that both f and f-1 are uniformly continuous. Show that (X,d) is complete if and only if (Y,d) is complete.
  6. [Cantor’s Intersection Theorem] Let (X,d) be a metric space and let {Fn} be a “decreasing” sequence of non-empty subsets of X satisfying Fn+1Fn for all n.
    1. Prove that if
      1. (X,d) is complete,
      2. each Fn is closed,
      3. diam(Fn)0,
      then nFn consists of exactly one point.
    2. Show that, if any of (i)-(iii) is omitted, then nFn may be empty.
    3. Conversely, prove that if for every decreasing sequence {Fn} of non-empty subsets satisfying (ii) and (iii), the intersection nFn is non-empty, then X is complete.
  7. Let (X,d) be a complete metric space and let f:X(0,) be a continuous function. Prove that there exists a point x* such that f(y)2f(x*) for all yB(x*,1f(x*)).
    (Hint: Arguing by contradiction show that there exists a sequence {xn} with the following properties: f(x1)>0, f(xn+1)>2f(xn) for all n1 and d(xn+1,xn)1f(xn). Then show that {xn} is Cauchy.)

Problem Set 5

  1. Let {fk} be a sequence of linear maps fk:nm which are not identically zero, that is, for every k there is x=xk such that fk(x)0. Show that there is x (not depending on k) such that fk(x)0 for all k.
  2. Let {fn} be a sequence of continuous functions fn: having the property that {fn(x)} is unbounded for all x. Prove that there is at least one xc such that {fn(x)} is unbounded.
  3. Let (X,d) be a complete metric space and let (Y,d) be a metric space. Let {fn} be a sequence of continuous functions from X to Y such that {fn(x)} converges for every xX. Prove that for every ε>0 there exist k and a non-empty open subset U of X such that d(fn(x),fm(x))<ε for all xU and all n,mk.
  4. Which of the following maps are contractions?
    1. f:, f(x)=e-x;
    2. f:[0,)[0,), f(x)=e-x;
    3. f:[0,)[0,), f(x)=e-ex;
    4. f:, f(x)=cosx;
    5. f:, f(x)=cos(cosx).
  5. Consider the map f:22 given by f(x,y)=110 (8x+8y,x+y), (x,y)2. Recall metrics d1((x1,y1),(x2,y2))=|x1-x2|+|y1-y2|, d2((x1,y1),(x2,y2))=[|x1-x2|2+|y1-y2|2]1/2 and d((x1,y1),x2,y2)=max{|x1-x2|,|y1-y2|}. Is f a contraction with respect to d1? d2? d?
    1. Consider X=(0,a] with the usual metric and f(x)=x2 for xX. Find values of a for which f is a contraction and show that f:XX does not have a fixed point.
    2. Consider X=[1,) with the usual metric and let f(x)=x+1x for xX. Show that f:XX and d(f(x),f(y))<d(x,y) for all xy, but f does not have a fixed point.
      Reconcile (a) and (b) with Banach fixed point theorem.
  6. 7. Let (X,d) be a complete metric space and f:XX be a function such that d(f(x),f(y)) αd(x,y) for all x,yB(x0,r0), where 0<α<1 and d(x0,f(x0))(1-α)·r0. Prove that f has a unique fixed point pB(x0,r0).
    1. Show that there is exactly one continuous function f:[0,1] which satisfies the equation [f(x)]3-ex [f(x)]2+ f(x)2=ex. (Hint: rewrite the equation as f(x)=ex+12f(x)1+f(x)2.)
    2. Consider C[0,a] with a<1 and T:C[0,a]C[0,a] given by (Tf)(t)=sint+ 0tf(s)ds,t [0,a]. Show that T is a contraction. What is the fixed point of T?
    3. Find all fC[0,π] which satisfy the equation 3f(t)=0tsin (t-s)f(s)ds.
    4. Let gC[0,1]. Show that there exists exactly one fC[0,1] which solves the equation f(x)+01 ex-y-1f(y) dy=g(x),for all x[0,1]. (Hint: Consider the metric d(f,h)=sup{e-x|f(x)-h(x)|:x[0,1]}.)

Problem Set 6

  1. On consider the metrics: d1(x,y) = |arctanx-arctany|, d2(x,y) = |x3-y3|. With which of these metrics is complete? If (,di) is not complete find its completion.
  2. Which of the following subsets of and 2 are compact? ( and 2 are considered with the usual metrics).
    1. A=[0,1]
    2. B={(x,y)2:x2+y2=1}
    3. C={(x,y)2:x2+y2<1}
    4. D={(x,y):|x|+|y|1}
    5. E={(x,y):x1and0y1/x}
  3. Prove that if A1,,Ak are compact subsets of a metric space (X,d), then i=1kAi is compact.
  4. Prove that if Ai is a compact subset of the metric space (Xi,di) for i=1,,k, then A1××Ak is a compact subset of X=X1××Xk with the product metric d.
  5. Let A be a non-empty compact subset of a metric space (X,d). Prove:
    1. If xX, then there exists aA such that d(x,A)=d(x,A);
    2. If AU and U is open, then there is ε>0 such that {xX:d(x,A)<ε}U.
    3. If B is closed and AB=, then d(A,B)>0.
    Hint: Recall that (x,y)d(x,y) is continuous from X×X[0,).
  6. Let f:X. Call a function f upper semicontinuous, abbreviated u.s.c., if for every r, {xX|f(x)<r} is open. Similarly, f is lower semicontinuous, abbreviated l.s.c., if for every r, {xX|f(x)>r} is open. Assume that X is compact. Show that every u.s.c. function assumes a maximum value and every l.s.c. function assumes a minimum value.
  7. Call a map f:XX a weak contraction if d(f(x),f(y))<d(x,y) for all xy. Prove that if X is compact and f is a weak contraction, then f has a unique fixed point.

The next problem gives a different construction of the completion of a metric space (X,d).

An equivalence relation on a set X is a relation having the following three properties:

  1. (Reflexivity) xx for every xX.
  2. (Symmetry) If xy, then yx.
  3. (Transitivity) If xy and yz, then xz.
The equivalence class determined by x, and denoted by [x], is defined by [x]={yX:yx}. We have [x]=[y] if and only if xy, and X is a disjoint union of these equivalence classes.

  1. Let (X,d) be a metric space and let X* be the set of Cauchy sequences x={xn} in (X,d). Define a relation in X* by declaring x={xn}y={yn} to mean d(xn,yn)0.
    1. Show that is an equivalence relation.
      Denote by [x] the equivalence class of xX*, and let X denote the set of these equivalence classes.
    2. Show that if x={xn} and y={yn}X*, then limnd(xn,yn) exists. Show that if x={xn}[x] and y={yn}[y], then limnd (xn,yn)= limnd (xn,yn). For [x],[y]X, define D([x],[y]) =limnd (xn,yn). Note that the definition of D is unambiguous in view of the above equality.
    3. Show that (X,D) is a complete metric space.
      Hint: Let [xn] be Cauchy in (X,D). Then xn={x1n,x2n,x3n,} is Cauchy in (X,d). So for every n, there exists kn such that d(xmn,xknn) <1/n for all mkn. Set xn={x1n,x2n,x3n,}. Then show that x is Cauchy in (X,d) and D([xn],[x])0.
    4. If xX, let φ(x) be the equivalence class of the constant sequence x=(x,x,x,). That is, φ(x)=[x]=[{x,x,x,}]. Show that φ:Xφ(X) is an isometry.
    5. Show that φ(X) is dense in (X,D).
      Hint: Let [x]X with x={x1,x2,x3,}. Denote by xn the constant sequence {xn,xn,xn,} and show that D([xn],[x])0.

Problem Set 7

  1. Consider the following spaces:
    1. with the metric d1(x,y)= |x-y|1+|x-y|;
    2. with the metric d2(x,y)= |arctanx-arctany|;
    3. with the metric d3(x,y)=0 if x=y and d(x,y)=1 if xy.
    Is (,di) compact?
  2. Use the Heine-Borel property to prove that if f:XY is a continuous mapping between metric spaces and X is compact then f is uniformly continuous.
  3. A family {Fi}iI is said to have the finite intersection property if for every finite subset J of I, iJFi. Show that X is compact if and only if for every family {Fi}iI of closed subsets of X having the finite intersection property, the intersection iIFi.
  4. Consider C[0,1] with the usual d metric. Let A= { fC[0,1]:0=f (0)f(t)f (1)=1for allt [0,1] } . Show that there is no finite 1/2-net for A.
  5. Show that if AX is totally bounded, then A is also totally bounded.
  6. Show that a metric space (X,d) is totally bounded if and only if every sequence {xn}X contains a Cauchy subsequence.
  7. Let X be a totally bounded metric space and Y a metric space. Assume that f:XY is a bijection. Show that if f and f-1 are uniformly continuous, then Y is totally bounded.
  8. [Lebesgue number lemma] Let (X,d) be a compact metric space and let {Ui}iI be an open covering of X. Prove that there exists δ>0 such that for every subset AX with diam(A)<δ there exists iI such that AUi.
    (δ is called a “Lebesgue number” for the covering.)
  9. Let (X,d) be a compact metric space. Assume that f:XX preserves distance, that is, d(f(x),f(y)) =d(x,y) for every x,yX. Show that f is a bijection.
    Hint: Assume that f(X)X. So there exists aX\f(X). Since f is continuous and X is compact, f(X) is compact. So d(a,f(X))=r>0. Consider a sequence xn=fn(a).

Problem Set 8

  1. Which of the following sets X are connected in 2?
    1. Let H={(x,y)2|xy=1andx,y>0}, L={(x,0)|x}, and X=HL;
    2. Let Cn={(x,y)2|(x-1/n)2+y2=1/n2} for n, and X=nCn.
  2. Show that if A is a connected subspace of a topological space (X,𝒯) and if ABA, then B is connected.
  3. If A and B are connected subsets of a topological space (X,𝒯) such that AB, then AB is connected.
  4. A point pX is called a cut point if X\{p} is disconnected. Show that the property of having a cut point is a topological property. (A property of a topological space is a topological property if it is preserved under homeomorphisms.)
  5. Show that no two of the intervals (a,b),(a,b], and [a,b] are homeomorphic.
  6. Show that and 2 are not homeomorphic (where and 2 are equipped with the usual topologies).
  7. Let S1={(x,y)2|x2+y2=1} be the unit circle in 2, and let f:S1 be a continuous function. Show that there exists xS1 such that f(x)=f(-x). [Hint: consider the function g:S1 where g(x)=f(x)-f(-x).]
  8. Let A be a countable set. Show that 2\A is path connected.
  9. Show that if A is an open connected subset of n, then A is path connected. [Hint: Fix a point x0A and consider the set U of all xA which can be joined to x0 by a path in A. Show that U and A\U are open.]
  10. A metric space (X,dX) is called a chain connected if for every pair x,y of points in X and every ε>0, there are finitely many points x=x0,x1,x2,,xn=y such that dX(xi+1,xi)<ε for i=0,1,,n-1. Prove that a compact, chain connected metric space is connected.

Problem Set 9

  1. Show that n becomes a real inner product space with X,Y=XTAY, where X,Yn are column vectors and A is a real symmetric matrix with positive eigenvalues. Similarly show that n is a complex inner product space with X,Y=XTBY, where B is a Hermitian matrix with positive eigenvalues. (Recall that a matrix B is Hermitian if BT=B, ie B is equal to the result of taking the complex conjugate transpose of B).
  2. Show that x=sup|x,y|y, over all y0 in any inner product space.
  3. Let H be the Hilbert space L2[-1,1]. Show that Gram Schmidt applied to the total set {1,t,t2,t3,} yields an orthonormal basis which is the sequence of Legendre polynomials given by Lk(t)=ck dkdtk (t2-1)k, k=1,2,3, where the ck are determined by requiring the polynomials to have unit length. In particular, show that the polynomials are orthogonal for any choice of ck. (You don’t need to compute the ck).
  4. Let W be a subspace of a Hilbert space H which admits an orthogonal projection P. Show;
    1. P2=P
    2. dist(x,W)=x-Px, ie Px is the closest point to x in W.
  5. Let A,B be subsets of a Hilbert space H. If A is the orthogonal complement of A, defined by A={xH:x,a=0,aA} then prove;
    1. A is a closed subspace of H
    2. AA{0}
    3. ABAB
    4. AA.
    5. If W is a subspace of H then W is closed if and only if W=W.
  6. Let S={e1,e2,} be a countable orthonormal basis for a separable Hilbert space H. Prove;
    1. x=nx,enen (Fourier series)
    2. x2=nx,en2 (Parseval's identity for norms)
    3. x,y=nx,eny,en (Parseval's identiy for inner products)
  7. If W,V are closed subspaces of a Hilbert space H and WV then show that W+V is closed.
  8. Let S be a subset of a Hilbert space H satisfying S={0}. Show that S is a total set in H, i.e S=H, ie the closure of the span of S is H.
  9. Let S be an arbitrary set. By l2(S) we mean the set of all functions f:S such that f(s)0 for countably many sS and such that the series {sS}|f(s)|2 converges. Define f,g={sS}f(s),g(s) for f,gl2(S). Prove that;
    1. l2(S) is a Hilbert space.
    2. l2=l2()
    3. Every Hilbert space with an orthonormal basis S is isometric to l2(S).
    (Hint: define functions fe:S by fe(e)=1, fe(e)=0 for e,eS, ee. Show that S={fe:eS} is an orthonormal basis for l2(S) and the bijection efe extends to an isometry Hl2(S).)

Problem Set 10

  1. Let R and L be the left and right shift operators in the normed space lp. So R(a1,a2,a3,)= (0,a1,a2,), L(a1,a2,a3,)= (a2,a3,) for all (a1,a2,)lp.
    1. Show that R,L are bounded linear operators.
    2. Show that R is injective but not surjective and L is surjective but not injective.
    3. Show that LR=I but RLI
    4. Show that Ln(x)0 for all xlp but Ln0.
    5. Find the norms L,R.
  2. For the case p=2 find the adjoints of the shift operators R,L:l2l2.
  3. Let a be a fixed element of a Hilbert space H. Prove that the mapping f(x)=x,a is a bounded linear functional with f=a.
  4. Prove the following facts about adjoints of bounded linear operators on Hilbert spaces.
    1. (T+S)*=T*+S*
    2. (TS)*=S*T*
    3. (λT)*=λT*
    4. T*T=T2
  5. Let Sn be a sequence of self adjoint operators on a Hilbert space H which converge pointwise to a bounded linear operator S. Show that S is self adjoint.
  6. If T is a positive operator on a Hilbert space H prove that Tn is positive for all n1.
  7. Prove that if T is a positive operator then every eigenvalue of T is non-negative.
  8. Let P be an orthogonal projection on a Hilbert space H. Prove that P is self adjoint, positive and I-P is positive.
  9. Let T:nm be a linear transformation with matrix [ajk] relative to the standard basis, so that the matrix is m×n. Suppose the norms in n,m are both the supremum norm. Show that T=maxjk|ajk|.
  10. Let T:nm be a linear transformation with matrix [ajk] relative to the standard basis, so that the matrix is m×n. Suppose the norms in n,m are both the l1 norms. Show that T=maxkk|ajk|.

Problem Set 11

  1. Let T be a bounded self adjoint compact operator on a Hilbert space H. If λ is a non zero complex number so that λI-T is a one-to-one mapping, prove that λI-T is onto and has a bounded inverse. (Hint: Use the spectral theorem).
  2. Let T be a bounded self adjoint compact operator on a Hilbert space H. If λ is a non zero complex number so that λI-T is an onto mapping, prove that λI-T is one-to-one and has a bounded inverse. (Hint: Show that if N is the null space for λI-T and R is the closure of the range of λI-T then N=R.)
  3. For the final example in the notes, check that the functions sn(t) are indeed eigenvectors for the Fredholm integral operator T associated to the Green’s function G with eigenvalues n2.
  4. Check that the functions sn(t) form an orthonormal set in L2[0,π].
  5. Check that the eigenvectors for L are the solutions to the Sturm Liouville system in the last example y+λy=0 on [0,π] with the boundary conditions y(0)=y(π)=0.

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