MAST30026 Assignments

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

Last update: 18 March 2014

Assignment 1

  1. Let A and B be bounded subsets of a metric space (X,d) such that AB. Show that diam(AB)diam (A)+diam(B). What can you say if A and B are disjoint?
  2. Let X=C[0,1] be the set of all continuous functions f:[0,1]. Recall that the supremum metric on X is defined by d(f,g)=sup { |f(x)-g(x)| :0x1 } and the L1 metric on X is defined by d1(f,g)= 01|f(x)-g(x)| dx. Consider the sequence {f1,f2,f3,} in X where fn(x)=nxn(1-x) for 0x1.
    1. Determine whether {fn} converges in (X,d1).
    2. Determine whether {fn} converges in (X,d).
    (You may use any standard results about limits of real sequences.)
  3. Let (X,dX) and (Y,dY) be metric spaces and let (X×Y,d) be the product metric space. Show that if AX and BY, then A×B= A×B.
  4. Let (X,d) be a metric space and let A be a non-empty subset of X. Recall that for each xX, the distance from x to A is d(x,A)=inf {d(x,a):aA}.
    1. Prove that A={xX:d(x,A)=0}.
    2. Prove that |d(x,A)-d(y,A)|d(x,y) for all x,yX. [Hint: first show that d(x,A)d(x,y)+d(y,A).]
    3. Deduce the function f:X defined by f(x)=d(x,A) is continuous.
    4. Show that if xA then U={yX:d(y,A)<d(x,A)} is an open set in X such that AU and xU.
  5. Determine whether the following sequences of functions converge uniformly.
    1. fn=x-nx2,x[0,1];
    2. gn=x-x2/n,x[0,1].
    3. gn=x-x2/n,x.
  6. Let X be the set of all real sequences with finitely many non-zero terms with the supremum metric: if x=(xi) and y=(yi) then d(x,y)=sup{|xi-yi|:i}.
    For each n, let xn=(1,1/2,1/3,,1/n,0,0,).
    1. Show that {xn} is a Cauchy sequence in X.
    2. Show that {xn} does not converge to a point in X. (So X is not complete.)
  7. Let X be a nonempty set and let (Y,d) be a complete metric space. Let f:XY be an injective function and define df(x,y)=d (f(x),f(y)) for x,yX.
    1. Explain briefly why df is a metric on X.
    2. Show that (X,df) is a complete metric space if f(X) is a closed subset of Y.
  8. Let f(x)=22+x forx0.
    1. Show that f defines a contraction mapping f:XX when X=[0,) is given the usual metric.
    2. Fix x00 and xn+1=f(xn) for all n0. Show that the sequence {xn} converges and find its limit with respect to the usual metric on .
  9. Let X be a complete normed vector space over . (Recall a norm · satisfies a+ba+b, a0 with a=0 if and only if a=0 and λa=|λ|a. We then define a metric by d(a,b)=a-b.) A sphere in X is a set S(a,r)= { xX:d(x,a)= x-a=r } where aX and r>0.
    1. Show that each sphere in X is nowhere dense.
    2. Show that there is no sequence of spheres {Sn} in X whose union is X.
    3. Give a geometric interpretation of the result in (b) when X=2 with the Euclidean norm.
    4. Show that the result of (b) does not hold in every complete metric space X.
  10. Let (X,d) and (Y,d) be metric spaces and let f,g:XY be continuous.
    1. Show that the set {xX:f(x)=g(x)} is a closed subset of X.
    2. Show that if f,g:X are continuous, then f-g is continuous and {xX:f(x)<g(x)} is open.

Assignment 2

  1. Let A={(x,y)2:x2+y2<1} and B={(x,y)2:(x-2)2+y2<1}.
    Determine whether X=AB, Y=AB and Z=AB are connected subsets of 2 with the usual topology. Explain your answers briefly.
  2. Let X be a connected topological space and let f:X be a continuous function, where has the usual topology. Show that if f takes only rational values, i.e. f(X), then f is a constant function.
  3. Show that X={(x,y)2:xy=0} is not homeomorphic to (with the usual topologies). [Hint: consider the effect of removing points from X and .]
  4. Prove that if X and Y are path connected, then X×Y is also path connected.
  5. Show that the following hold for subsets of a topological space X;
    • if subsets A,B are path connected and AB then AB is path connected.
    • Show that every point of X is contained in a unique path component, which can be defined as the largest path connected subset of X containing this point.
    • Give examples to show that the path components need not be open or closed.
    • Prove that if X is locally path connected, i.e every point of x is contained in an open set U which is path connected, then every path component is open.
    • Conclude that if X is locally path connected, then the path compo- nents coincide with the connected components.
  6. Let X=C1[0,1], Y=C[0,1] so that functions in X are continuously differentiable and functions in Y are continuous. Define norms f=supt|f(t)| on Y and f0=f+f on X, where f=dfdt. Let D:XY be the differentiation operator Df=dfdt.
    1. Show that D:(X,·0)(Y,·) is a bounded linear operator with D=1.
    2. Show that D:(X,·)(Y,·) is an unbounded linear operator. (Hint: Consider the sequence of elements tn in X).
  7. Let {a1,a2,} be a bounded sequence of complex numbers. Define an operator T:l2l2 by; T(b1,b2,)= (0,a1b1,a2b2,).
    1. Show that T is a bounded linear operator and find T.
    2. Compute the adjoint operator T*.
    3. Show that A*AAA* whenever A0.
    4. Find the eigenvalues of A*.
  8. Let [aij] be an infinite complex matrix, i,j=1,2,, so that cj=i|aij| converges for every j and the sequence cj is bounded with c=supjcj. Show that the operator T:l1l1 defined by T(b1,b2,)=(ja1jbj,ja2jbj,) is a bounded linear operator and Tc. Deduce that T=c.

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