Problems for Metric and Hilbert Spaces February 2014

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 27 February 2014

Homework

  1. Define the standard metric on and show that , with this metric, is a metric space.
  2. Let d be the standard metric on . Show that is a metric subspace of (,d).
  3. Let X be a set. Define the standard metric on X and show that X, with this metric, is a metric space.
  4. Let (X1,d1),,(Xn,dn) be metric spaces. Define the product metric d on X1××Xn and show that (X1××Xn,d) is a metric space.
  5. Let (X,) be a normed vector space. Define the standard metric on X and show that X, with this metric, is a metric space.
  6. Define the standard metric on n and show that n, with this metric, is a metric space.
  7. Define the standard norm on n and show that n, with this norm, is a normed vector space.
  8. Define the norm p on n and show that (n,p) is a normed vector space.
  9. Let X be a nonempty set. Define the set of bounded functions B(X,) and the sup norm on B(X,). Show that B(X,), with this norm, is a normed vector space.
  10. Let a,b with a<b. Define the set of continuous functions C([a,b],) and the L1-norm on C([a,b],). Show that C([a,b],), with this norm, is a normed vector space.
  11. Let a,b with a<b. Show that the set Cbd([a,b],) of bounded continuous functions is a metric subspace of C([a,b],) with the L1-norm.
  12. Let X be a metric space and let x1,x2, be a sequence in X. Show that limnxn is unique, if it exists.
  13. Let (X1,d1),,(X,d) be metric spaces. Show that a sequence xn=(xn(1),,xn()) in X1××X converges if and only if each of the sequences xn(i) (in Xi) converges.
  14. Let (X,d) be a metric space. Show that the metric d:X×X given by d(x,y)= d(x,y) 1+d(x,y) is equivalent to d.
  15. Let (X,d) be a metric space. Show that (X,d) is a bounded metric space, where d(x,y)= d(x,y) 1+d(x,y) .
  16. Give an example of X and two metrics d and d on X such that d is equivalent to d and (X,d) is not bounded and (X,d) is bounded.
  17. Let (X1,d1),,(X,d) be metric spaces and let (X1××X,d) be the product metric space. Let σ:(X1××X)×(X1××X) given by σ(x,y)=max { di(xi,yi) |1i } . Show that σ is a metric on X1××X and d is equivalent to σ.
  18. Let (X1,d1),,(X,d) be metric spaces and let (X1××X,d) be the product metric space. Let ρ:(X1××X)×(X1××X) be given by ρ(x,y)= ( i=1di (xi,yi)2 ) 12 . Show that ρ is a metric on X1××X and d is equivalent to ρ.
  19. Let X be a set and let d and d be metrics on X. Show that d and d are equivalent if and only if d and d satisfy the condition if x,yX then there exist k,k such that d(x,y)kd(x,y) kd(x,y).
  20. Let (X,d) be a metric space. Define the metric space topology on X and show that it is a topology on X.
  21. Let X be a set and let d be the discrete metric on X. Determine which subsets of X are in the metric space topology on X.
  22. Give two metrics d and d on such that is open in the metric space topology on (,d) and is not open in the metric space topology on (,d).
  23. Let X be a topological space and let E be a subset of X. Let E be the interior of E. Show that E is open if and only if E=E.
  24. Let X be a topological space and let E be a subset of X. Let E be the interior of E. Show that E is the set of interior points of E.
  25. Let X be a topological space and let xX. Consider the following definitions of "neighborhood of x":
    1. A neighborhood of x is a set NX such that xN.
    2. A neighborhood of x is a set VX such that there exists an open set U of X with xUV.
    Show that these two definitions of "neighborhood of x" are equivalent.
  26. Let X be a topological space and let E be a subset of X. Let E be the closure of E. Show that E is closed if and only if E=E.
  27. Let X be a topological space and let E be a subset of X. Let xX. Show that x is a closed point of E if and only if there exists a sequence x1,x2, of points in E such that limnxn=x.
  28. Let (X,d) be a metric space and let xX and r>0. Show that the closed ball B(x,r)= {yX|d(x,y)r} is a closed set in the metric space topology on X.
  29. Give an example of a metric space (X,d) and a point xX such that B(x,1)B(x,1).
  30. Let (X,d) be a metric space and let xX and r>0. Show that B(x,r)B(x,r).
  31. In with the usual topology give an example of
    1. a set A which is both open and closed,
    2. a set B which is open and not closed,
    3. a set C which is closed and not open,
    4. a set D which is not open and not closed.
  32. Let X= with the usual topology. Show that
    1. [0,1) is not open and not closed,
    2. is not open and not closed.
  33. Let X be a set with the discrete metric d. Show that every subset of X is both open and closed (in the metric space topology on X).
  34. Let X be a set and let 𝒞 be a collection of subsets of X. Show that 𝒞 is the set of closed sets for a topology on X if and only if 𝒞 satisfies
    1. finite unions of elements of 𝒞 are in 𝒞,
    2. Arbitrary intersections of elements of 𝒞 are in 𝒞,
    3. 𝒞 and X𝒞.
  35. Let X be a topological space and let YX with the subspace topology. Show that
    1. BY is open in Y if and only if B=YA for some set AX which is open in X.
    2. BY is closed in Y if and only if there exists FX closed in X such that B=YF.
  36. Let X1,,X be topological spaces and let X1××X have the product topology. Show that
    1. If A1X1,,AX are open then A1××AX1××X is open.
    2. If F1X1,,FX are closed then F1××FX1××X is closed.
  37. Let (X1,d1),,(X,d) be metric spaces and let d be the product metric on X1××X. Show that the metric space topology on (X1××X,d) is the product topology for X1××X, where X1,,X have the metric space topology.
  38. Let X be a topological space and let AX. Show that if xX satisfies if r>0 then B(x,r)A and B(x,r)Ac then xA.
  39. Let X be a topological space and let AX. Show that A is a closed subset of X.
  40. Let X= with the usual topology.
    1. Determine (with proof) ([0,1]).
    2. Determine (with proof, of course).
  41. Let (X,d) be a metric space. Let xX. Show that {x}X is closed (in the metric space topology on X).
  42. Let (X,d) be a metric space and let xX. Show that x is isolated if and only if there exists ε>0 such that B(x,ε)={x}.
  43. Let X= with the usual topology. Show that
    1. >0 is a discrete set in ,
    2. {1n|n>0} is a discrete set in .
  44. Let X be a discrete topological space. Show that every subset of X is both open and closed.
  45. Let X be a topological space. Show that X is discrete if and only if the only convergent sequences are those which are eventually constant.
  46. Let X= with the usual topology.
    1. Show that is dense in .
    2. Show that c is dense in .
    3. Show that >0 is nowhere dense in .
    4. Show that is nowhere dense in .
    5. Show that is nowhere dense in 2.
  47. Let C be the Cantor set in , where has the usual topology.
    1. Show that C is closed in .
    2. Show that C does not contain any interval in .
    3. Show that C has nonempty interior.
    4. Show that C is nowhere dense in .
  48. Let (X,d) and (Y,ρ) be metric spaces and let f:XY be a function. Let aX. Show that f is continuous at a if and only if f satisfies: if ε>0 then there exists δ>0 such that
    if xX and d(x,a)<δ then ρ(f(x),f(a))<ε.
  49. Let X and Y be topological spaces and let f:XY be a function. Show that f is continuous if and only if f satisfies: if aX then f is continuous at a.
  50. Let X and Y be metric spaces and let f:XY be a function. Let aX. Show that f is continuous at a if and only if f satisfies: if ε>0 then there exists δ>0 such that f(B(a,δ)) B(f(a),ε).
  51. Let X and Y be metric spaces and let f:XY be a function. Let aX. Show that f is continuous at a if and only if f satisfies if x1,x2, is a sequence in X and limnxn=x0 then limnf(xn)=f(x0).
  52. Let X and Y be metric spaces and let f:XY be a function. Let aX. Show that f is continuous at a if and only if f satisfies: if x1,x2, is a convergent sequence in X then limnf(xn)=f(limnxn).
  53. Let X and Y be topological spaces. Let f:XY be a function. Show that f is continuous if and only if f satisfies: if FY is closed then f-1(F) is closed in X.
  54. Let X,Y and Z be topological spaces and let f:XY and g:YZ be continuous functions. Show that gf is a continuous function.
  55. Let X,Y be topological spaces and let f:XY be a continuous function. Let AX. Show that the restriction of f to A, f|A:AY is continuous.
  56. Let (X,d),(Y1,ρ1) and (Y2,ρ2) be metric spaces. Let f:XY1 and g:XY2 be functions. Define h:XY1×Y2 by h(x)=(f(x),g(x)). Let aX. Show that h is continuous at a if and only if f and g are continuous at a. X Δ X×X f×g Y1×Y2 x (x,x) (f(x),g(x))
  57. Let (X,d),(Y1,ρ1) and (Y2,ρ2) be metric spaces. Let f:XY1 and g:XY2 be functions. Define h:XY1×Y2 by h(x)=(f(x),g(x)). Let aX. Show that h is continuous if and only if f and g are continuous.
  58. Let (X,d) be a metric space and let f:X and g:X be continuous functions.
    1. Show that f+g is continuous.
    2. Show that f·g is continuous.
    3. Show that f-g is continuous.
    4. Show that if g satisfies: if xX then g(x)0 then f/g is continuous.
  59. Let X be a topological space and let f:X and g:X be continuous functions.
    1. Show that f+g is continuous.
    2. Show that f·g is continuous.
    3. Show that f-g is continuous.
    4. Show that if g satisfies: if xX then g(x)0 then f/g is continuous.
  60. Let (X,d) be a metric space. Show that d:X×X is continuous.
  61. Let f:× be given by f(x,y)= { xyx2+y2, if(x,y) (0,0) 0, if(x,y)= (0,0). If a let a: be given by a(y)=f(a,y). If b let rb: be given by rb(x)=f(x,b).
    1. Let a. Show that a: is continuous.
    2. Let b. Show that rb: is continuous.
    3. Show that f is not continuous at (0,0).
  62. Give an example of metric spaces X,Y and Z and a function f:X×Y such that
    1. if xX then x: Y Z y f(x,y) is continuous,
    2. if yY then ry: X Z x f(x,y) is continuous,
    3. f:X×YZ is not continuous.
  63. Let X be a topological space and let AX and BX be closed subsets of X such that X=AB. Let Y be a topological space and let f:AY and g:BY be continuous functions such that if xAB then f(x)=g(x). Define h:XY by h(x)= { f(x), ifxA, g(x), ifxB. Show that h:XY is continuous.
  64. Show that the function f: given by f(x)= x1+x2, is uniformly continuous.
  65. Show that the function f: given by f(x)=x2, is not uniformly continuous.
  66. Let (X,d) and (Y,ρ) be metric spaces and let f:XY be a function. Show that if f is uniformly continuous then f is continous.
  67. Let (X,d) and (Y,ρ) be metric spaces. Let {fk} be a sequence of functions fk:XY and let f:XY be a function. Show that {fk} converges uniformly to f if and only if sup{ρ(fk(x),f(x))|xX}0.
  68. Let {fk} be a sequence of continuous functions from a metric space (X,d) to a metric space (Y,ρ). Suppose that {fk} converges uniformly to f:XY. Show that f:XY is continuous.
  69. Let (X,d) be a metric space and let x1,x2, be a sequence in X. Show that if (x1,x2,) is a Cauchy sequence then {x1,x2,} is bounded.
  70. Let (X,d) be a metric space and let (x1,x2,) be a sequence in X. Show that if (x1,x2,) converges then (x1,x2,) is a Cauchy sequence.
  71. Let (X,d) be a metric space and let (x1,x2,) be a sequence in X. Show that if (x1,x2,) is a Cauchy sequence and contains a convergent subsequence then (x1,x2,) converges.
  72. Give an example of a metric space (X,d) and a Cauchy sequence (x1,x2,) in X that does not converge.
  73. Give an example of a metric space (X,d) that is not complete.
  74. Show that with the usual metric is a complete metric space.
  75. Let (X,d) be a complete metric space. Let YX be a subspace of X. Show that if Y is closed then (Y,d) is complete.
  76. Give an example of a metric space (X,d) and a subspace YX such that (X,d) is a complete metric space and (Y,d) is not complete.
  77. Let (X,d) be a metric space and let YX be a subspace of X. Show that if (Y,d) is complete then Y is a closed subset of X.
  78. Let (X1,d1),,(X,d) be metric spaces and let (X1××X,d) be the product metric space. Show that if (X1,d1),,(X,d) are complete then (X1××X,d) is complete.
  79. Let (X,d) and (Y,d) be metric spaces and let Cb(X,Y) be the set of bounded continuous functions f:XY with the metric given by ρ(f,g)=sup { d(f(x),g(x)) |xX } . Show that if (Y,d) is complete then (Cb(X,y),ρ) is a complete metric space.
  80. Let (X,d) and (Y,d) be metric spaces and let Cb(X,Y) be the set of bounded continuous functions f:XY with the metric ρ:Cb(X,Y)×Cb(X,Y)>0 given by ρ(f,g)=sup { d(f(x),g(x)) |xX } . Show that (Cb(X,Y),ρ) is a metric space.
  81. Let (X,d) be a metric space and let UX and VX. Show that if U and V are open and dense then UV is open and dense.
  82. Let X= with the usual metric and let U= and V=c. Show that U and V are dense and UB=.
  83. Let X= with the usual metric and let ={q1,q2,q3,} be an enumeration of . For n>0 let Qn=\{qn}.
    1. Show that if n>0 then Qn is open and dense.
    2. Show that n>0Qn=.
  84. Let (X,d) be a complete metric space and let {U1,U2,U3,} be a sequence of open and dense subsets of X. show that n>0Un is dense in X.
  85. Let (X,d) be a complete metric space and let {F1,F2,F3,} be a sequence of nowhere dense subsets of X. Show that n>0Fn has empty interior.
  86. Show that , with the standard topology, cannot be be written as a countable union of nowhere dense sets.
  87. Let X=, with the standard topology. Let ={q1,q2,q3,} be an enumeration of .
    1. Show that {qn} is nowhere dense.
    2. Determine the interior of n>0{qn}.
  88. Let (X,d) be a complete metric space and let {f1,f2,f3,} be a sequence of continuous functions fn:X,for n>0. Assume that if xX then {f1(x),f2(x),} is bounded in X. Show that there exists and open set UX such that there exists M>0 such that
    if xU and n>0 then |fn(x)|M.
  89. Show that the completion of (0,1) with the usual metric is [0,1] with the usual metric.
  90. Let (X,d) and (Y,ρ) be metric spaces and let f:XY be an isometry. Show that f is injective.
  91. Give an example of an isometry f:XY that is not surjective.
  92. Let (X,d) be a metric space. Show that a completion of (X,d) exists.
  93. Let (X,d) be a metric space. Show that the completion of (X,d) is unique (if it exists).
  94. Let (X,d) be a metric space. Let ((X1,d1),φ1) and ((X2,d2),φ2) be completions of (X,d). Show that there is a surjective isometry f:X1X2 such that fφ1=φ2.
  95. Let X={0,1} and let 𝒯={,X,{0}}.
    1. Show 𝒯 is a topology on X.
    2. Show that there does not exist a metric d:X×X0 such that 𝒯 is the metric space topology of (X,d).
  96. Let X be a complete metric space and let f:XX be a contraction. Show that f has a unique fixed point.
  97. Let α with 0<α<1. Let X be a complete metric space and let f:XX be a α-contraction. Let xX, x0=x and xn+1=f(xn), for n0.
    1. Show that the sequence x0,x1,x2, converges in X.
    Let p=limnxn.
    1. Show that d(x,p)d(x,f(x))1-α.
    2. Show that f(p)=p.
  98. Let U be an open subset of 2. Let f:U be a continuous function which satisfies the Lipschitz condition with respect to the second variable: There exists α>0 such that
    if (x,y1),(x,y2)U then |f(x,x1)-f(x,y2)|α|y1,y2|.
    Show that if (x0,y0)U then there exists δ>0 such that y(x)=f(x,y(x)) has a unique solution y:[x0-δ,x0+δ] such that y(x0)=y0.
  99. Let (X,) be a normed vector space. Show that (X,) is complete if and only if every norm absolutely convergent series is convergent in X.
  100. Let S be the set of linear combinations of step functions f:k. Let f=|f| andd(f,g)= f-g for f,gS.
    1. show that :S0 is not a norm on S.
    2. Show that d:S×S0 is not a metric on S.
  101. Let S be the set of linear combinations of step functions f:k. Let i>0fi be a series in S which is norm absolutely convergent. Show that there exists a full set in k on which i0fi converges.
  102. Let S be the set of linear combinations of step functions f:k. Let n>0fk be a series in S which is norm absolutely convergent. Show that n>0fn=0 almost everywhere if and only if the limit of the norms of the partial sums of fn converge to 0.
  103. Let L be the set of functions which are equal almost everywhere to limits of norm absolutely convergent series in S, where S is the set of linear combinations of step functions f:k. Define f=fand d(f,g)=f-g for f,gL.
    1. Show that :L0 is a norm on L.
    2. Show that d:L×L0 is a metric on L.
  104. Let I be a closed and bounded interval in . Let x1,x2,x3,, be a sequence in I. Show that there exists a subsequence xn1,xn2,xn3, of x1,x2,x3, such that xn1,xn2,xn3, converges in I.
  105. Let X be a compact topological space. Let C be a closed subset of X. Show that C is compact.
  106. Let X be a metric space and let E be a compact subset of X. Show that E is closed and bounded.
  107. Let C([0,1],)={f:[0,1]|fis continuous} and let d(f,g)=sup { |f(x)-g(x)| |x[0,1] } .
    1. Show that d:C([0,1],)×C([0,1],) is a metric on C([0,1],).
    2. Let A=B1(0)={fC([0,1],)|d(f,0)1}. Show that A is closed and bounded.
    3. Show that A is not compact.
  108. Let K. Show that K is compact if and only if K is closed and bounded.
  109. Let (X,d) and (Y,d) be metric spaces and let f:XY be a continuous function. Let K be a compact subset of X. Show that f(K) is compact in Y.
  110. Let X be a compact metric space. Let f:X be a continuous function. Show that f attains a maximum and a minimum value.
  111. Let X be a compact metric space. Let f:XY be a continuous function. Show that f is uniformly continuous.
  112. Let X be a set with the discrete metric. Show that X is compact if and only if X is finite.
  113. Let X be a metric space and let AX. Show that if A is totally bounded then A is bounded.
  114. Let X= with metric given by d(x,y)=min {|x-y|,1}.
    1. Show that X is bounded.
    2. Show that X is not totally bounded.
  115. Let X be a metric space and let AX. Show that the following are equivalent:
    1. Every sequence in A has a convergent subsequence.
    2. A is complete and totally bounded.
    3. Every open cover of A has a finite subcover.
  116. Let X be a topological space. Show that X is compact if and only if X satisfies if 𝒞 is a collection of closed sets such that if >0 and C1,,C𝒞 then C1C then C𝒞C.
  117. Let X be a topological space and let KX. Assume X is compact. Show that if K is closed then K is compact.
  118. Let X be a topological space and let KX. Assume X is Hausdorff. Show that if K is compact then K is closed.
  119. Show that a compact Hausdorff space is normal.
  120. Let X and Y be topological spaces and let f:XY be a continuous function. Let KX. Show that if K is compact then f(K) is compact.
  121. Let X and Y be topological spaces and let f:XY be a continuous function. Assume f is a bijection, X is compact and Y is Hausdorff. Show that the inverse function f-1:YX is continuous.
  122. Let X=[0,2π) and Y=S1={(x,y)2|x2+y2=1}. Let f:[0,2π)S1 be given by f(x)=(cosx,sinx).
    1. Show that f is continuous.
    2. Show that f is a bijection.
    3. Show that f-1:S1[0,2π) is not continuous.
    4. Why does this not contradict the previous problem?
  123. Let X be a set with Card(X)>1.
    1. Show that X with the discrete topology is disconnected.
    2. Show that X with the indiscrete topology is connected.
  124. Let X1 and X2 be the subspaces of given by X1-\{0} and X2=. Show that X1 and X2 are disconnected.
  125. Let Y={0,1} with the discrete topology. Let X be a topological space. Show that X is connected if and only if every continuous function f:XY is constant.
  126. Let X and Y be topological spaces and let f:XY be a continuous function. Let EX. Show that if E is connected then f(E) is connected.
  127. Let X be a connected topological space and let AX. Show that if A is connected then the closure of A, A, is connected.
  128. Let A=(-,0) and B=(0,) as subsets of . Show that A is connected, B is connected and AB is not connected.
  129. Let X be a topological space. Let 𝒮 be a collection of subsets of X such that A𝒮A. Show that A𝒮A is connected.
  130. Let X be a topological space such that if x,yX then there exists AX such that xA, yA and A is connected. Show that X is connected.
  131. Let X be a topological space. For xX let Cx be the connected component containing x.
    1. Let yX. Show that Cy is connected and closed.
    2. Show that the connected components of X partition X.
  132. Let X be a set with the discrete topology. Determine (with proof) the connected components of X.
  133. Show that a subset of is connected if and only if it is an interval.
  134. Carefully state the Intermediate Value Theorem.
  135. State and prove the Intermediate Value Theorem.
  136. Let X be a connected topological space and let f:X be a continuous function. Show that if x,yX and r such that f(x)rf(y) then there exists cX such that f(c)=r.
  137. Let X be a topological space. Show that if X is path connected then X is connected.
  138. Let X={(t,sin(πt))|t(0,2]}.
    1. Let φ:2 be given by φ(x,y)=x. Show that φ:X(0,2] is a homeomorphism.
    2. Show that X is connected.
    3. Show that X is connected.
    4. Show that X is not path connected.
  139. Let p1. Let V=n with (a1,,an)p= ( |a1|p ++ |an|p ) 1p . Show that (V,p) is a Banach space.
  140. Let V=n with (a1,,an)= sup{|a1|,|a2|,,|an|}. Show that (V,) is a Banach space.
  141. Let p be the vector space of sequences (a1,a2,) in such that n>0|an|p<. Let (a1,a2,)q= (n>0|an|p)1p. Show that (p,p) is a Banach space.
  142. Let be the vector space of bounded sequences (a1,a2,) in with (a1,a2,)=sup {|ai||i>0}. Show that (,) is a Banach space.
  143. Let X be a topological space. Let F= or . Let Cb(X,F) be the vector space of bounded continuous functions with f=sup {|f(x)||xX}. Show that (Cb(X,F),) is a Banach space.
  144. Let (V,) be a normed vector space. Show that (V,) is a Banach space if and only if every norm absolutely convergent series is convergent.
  145. Let (V,) be a normed vector space. Show that a Schauder basis of V is a total set.
  146. Let (V,) be a normed vector space. Show that if V has a Schauder basis then V is separable.
  147. Let (V,) be a normed vector space. Show that there exists a countable dense set in V is and only if there exists a countable total set in V.
  148. Let ei=(0,0,,0,1,0,0,) with 1 in the ith entry. Show that {e1,e2,e3,} is a Schauder basis of p.
  149. Show that is not separable.
  150. Show that does not have a Schauder basis.
  151. Let Cb([0,1],) be the vector space of bounded continuous functions on [0,1].
    1. Show that the set of polynomials is dense in the space of continuous functions on [0,1] with the supremum norm.
    2. Show that the polynomials with rational coefficients form a countable dense set in Cb([0,1],).
    3. Show that Cb([0,1],) is separable.
  152. Let (V,) be a Banach space.
    1. Show that if dim(V)< then the closed unit ball in V is compact.
    2. Show that if V is infinite dimensional then the closed unit ball in V is not compact.
  153. Let V be a finite dimensional vector space. Let 1 and 2 be norms on V. Show that 1 and 2 are equivalent.
  154. Let (V,) be a normed vector space. Show that if V is finite dimensional then the closed unit ball in V is compact.
  155. Let (V,) be an infinite dimensional Banach space. Construct a sequence (e1,e2,) of unit vectors in V such that if i,j>0 and ij then d(ei,ej)>12.
  156. Let V=n with (x1,x2,,xn), (y1,y2,,yn) =i=1nxi yi. show that (V,) is a Hilbert space.
  157. Let 2 be the set of sequences (a1,a2,) in such that i>0|ai|2<. Let (x1,x2,), (y1,y2,) =i>0 xiyi. Show that (2,) is a Hilbert space.
  158. Carefully define the space L2([a,b]) and show that if f,g= abf(t) g(t)dt then (L2([a,b]),,) is a Hilbert space.
  159. Let (V,) be an inner product space and let x=x,x for xV. Show that if x,yV then x+y2+ x-y2= 2x2+ 2y2.
  160. Let (V,) be a normed vector space. Define x,y=14 ( x+y2- x-y2+i x+iy2-i x-iy2 ) for x,yV. Show that if satisfies if x,yV then x+y2+ x-y2= 2x2+ 2y2 then is a norm on V.
  161. Let (V,) be an inner product space. Show that the Gram-Schmidt process produces an orthonormal basis of V.
  162. Let (V,) be an inner product space. Let W be a vector subspace of V. Show that if W admits an orthogonal projection P then P is unique.
  163. Let (V,) be a Hilbert space. Let W be a vector subspace of V. Show that if there is an orthogonal projection P onto W then W is closed.
  164. Let (V,) be a Hilbert space. Let (a1,a2,) be an orthonormal sequence in V. Let W=span{a1,a2,} and let M=W be the closure of W. Show that P:VV given by P(x)=n>0 x,anan is an orthonormal projection onto M.
  165. Let (V,) be a Hilbert space. Let (a1,a2,) be an orthonormal sequence in V. Let xV. Show that P(x)=n>0 x,anan is independent of the order of the terms in the sum.
  166. Let (V,) be a Hilbert space. Let (a1,a2,) be an orthonormal sequence in V. Let xV. Show that n>0 |x,an|2 x2.
  167. Let (V,) be a separable Hilbert space. Show that V has a Schauder basis.
  168. Let (V,) be a Hilbert space. Assume that V has a countable orthonormal set {a1,a2,} which is a total set. Show that {a1,a2,} is a Schauder basis for V.
  169. Show that the functions em(t)= 12πeimt, form, form an orthonormal basis of L2([0,2π]).
  170. Let (V,) be a Hilbert space and let W be a closed subspace of V. Show that V=WW.
  171. Let V and W be normed vector spaces and let T:VW be a linear transformation. Show that if V is finite dimensional then T is bounded.
  172. Let V=C([0,1]) be the vector space of continuous functions f:[0,1] with norm given by f=01 |f(t)|dt. Let T:V be given by T(f)=f(0).
    1. Show that V is infinite dimensional.
    2. Show that T is not bounded.
  173. Let V,W be normed vector spaces and let T:VW be a linear transformation. Show that if T is continuous then T is bounded.
  174. Let V,W be normed vector spaces and let T:VW be a linear transformation. Show that if T is bounded then T is uniformly continuous.
  175. Let V,W be normed vector spaces. Show the identity operator idV:VV has operator norm 1 and the zero operator 0:VW has operator norm 0.
  176. Let be the vector space of bounded sequences (a1,a2,) in with norm given by (a1,a2,)=sup {|a1|,|a2|,}. Let (λ1,λ2,) be a bounded sequence in . Define T: by T(a1,a2,)= (λ1a1,λ2a2,).
    1. Show that T is a well defined linear transformation.
    2. Show that T=sup {|λ1|,|λ2|,}.
  177. Let a,b with a<b. Let k:[a,b]×[a,b] be a continuous function. Let X={f:[a,b]|fis continuous} with the supremum norm. Define T:XX by (Tf)(t)=abk(t,s)x(s)ds.
    1. Show that X is a Banach space.
    2. Show that if fX then TfX.
    3. Show that T is a bounded linear transformation.
  178. Let V,W be normed vector spaces. Let B(V,W) be the vector space of bounded linear operators with the operator norm. Show that if W is a Banach space then B(V,W) is a Banach space.
  179. Let a,b with a<b. Let C([a,b]) be the vector space of continuous functions f:[a,b] with the supremum norm. Define T:C([a,b]) by Tf=abf(t)dt Show that the operator norm of T is T=b-a.
  180. Let H1 and H2 be Hilbert spaces and let T:H1H2 be a bounded linear transformation.
    1. Show that there exists a unique function T*:H2H1 such if xH1 and yH2 then Tx,y2= x,T*y1.
    2. Show that T* is a linear transformation.
    3. Show that T* is bounded.
    4. Show that T*=T.
  181. Let H be a Hilbert space and let f:H be a bounded linear functional. Show that there exists a unique aH such that if xH then f(x)=x,a.
  182. Let H1 and H2 be Hilbert spaces and let T:H1H2 be a bounded linear transformation. Show that T**=T.
  183. Let a,b with a<b. Let C([a,b]) be the Banach space of continuous functions f:[a,b] with the supremum norm. Let t0[a,b]. Define A:C([a,b]) by Af=f(t0). Show that A is a bounded linear functional with A=1.
  184. Let T:mn be a linear transformation. Let A be the matrix of T and let A*=At.
    1. Show that the matrix of T* is A*.
    2. Show that T=γ, where γ is the largest eigenvalue of A*A.
  185. Let V and W be normed vector spaces and let T:VW be a bounded linear operator. Show that T*T is self adjoint and positive.
  186. Let p1 and let q be defined by 1p+1q=1. Show that the dual of the Banach space p is q.
  187. Let p1. Show that p is a reflexive Banach space.
  188. Let X be a normed vector space and let B={xX|x1}. Let T:XX be a compact operator. Show that T(B) is compact.
  189. Let X be a normed vector space and let A be a bounded subset of X. Let T:XX be a compact operator. Show that T(A) is compact.
  190. Let X be a finite dimensional normed vector space and let T:XX be a linear transformation. Show that T is a compact operator.
  191. Let X=([a,b]) be the space of continuous functions f:[a,b] with the supremum norm. Let k:[a,b]×[a,b] be a continuous function and define T:XX by (Tf)(t)= abk(t,s)f(s)ds Show that T is a compact operator.
  192. Let H be a Hilbert space and let T:HH be a bounded self adjoint operator. Show that T=sup { |Tx,x| |xH,x=1 } .
  193. Let H be a Hilbert space and let T:HH be a nonzero compact self adjoint operator.
    1. Show that there exists an eigenvalue λ of T such that |λ|=T.
    2. Show that if v is an eigenvector of T with eigenvalue λ such that |λ|=T then v is a solution of the extremal problem max { Tu,u |uH,u =1 } .
  194. Let H be a Hilbert space and let T:HH be a nonzero compact self adjoint operator. Show that there exists an orthonormal basis of eigenvectors for H.
  195. Let H be a Hilbert space and let T:HH be a nonzero compact self adjoint operator. Let Λ be the set of eigenvalues of T. If μΛ let P(μ) be the orthogonal projection onto the subspace Xμ of eigenvectors with eigenvalue μ. Show that if xH then Tx=μΛμP(μ)x.
  196. Let V and W be Banach spaces. Let V be the dual of V and let W be the dual of W. Let T:VW be a bounded linear operator. Define T*:WV by T*f=fT. Show that T* is a well defined bounded linear operator.
  197. Let V be a Banach space and let V be the dual of the dual of V. Define φ:VV by (φ(x))(f)= f(x),forf V. Show that φ is injective.
  198. Let V and W be reflexive Banach spaces and let T:VW be a bounded linear operator.
    1. Show that T is transformed to T** by the isomorphisms VV and WW.
    2. Show that T*=T.
    3. Show that if V and W are Hilbert spaces then T is transformed to T* by the natural isomorphisms VV and WW.
  199. Let H be an infinite dimensional Hilbert space. Let T:HH be a bounded self adjoint compact operator. Show that the eigenvalues of T form a sequence converging to 0 and every eigenspace for T is finite dimensional.
  200. Let a,b with a<b. Let λ and let p:[a,b]>0 and q:[a,b] with pC1([a,b]) and qC2([a,b]). Let a1,a2,b1,b2 with (a1,a2)(0,0) and (b1,b2)(0,0). Let L:C2([a,b])C([a,b]) be given by Ly=(-py)+qy. Let u,vC2([a,b]) such that Lu=0,Lv=0, a1u(a)+a2u (a)=0,and b1v(b)+ b2v(b)=0. Let G:[a,b]×[a,b] be given by G(s,t)= { v(t)u(s), ifst u(t)v(s), ifts. Define T:L2([a,b])L2([a,b]) by (Tf)(t)= abG(t,s)f(s) ds,fort[a,b].
    1. Show that the eigenvalues of T are nonzero and each eigenvector f satisfies a1f(a)+a2f(a)=0 and b1f(b)+b2f(b)=0.
    2. Show that f is an eigenvector of T with eigenvalue μ if and only if f is an eigenvector of L with eigenvalue 1μ.
    3. Show that L has a sequence of eigenvalues λ, each eigenspace of L is one dimensional and there is an orthonormal basis of L2([a,b]) of eigenvectors of L.
  201. Let G:[0,π]×[0,π] be given by G(t,s)= { t-stπ, ifst, s-stπ, ifst and let T:L2([0,π])L2([0,π]) be given by (Tf)(t)=0π G(t,s)f(s)ds fort[0,π].
    1. Show that T has eigenvalues λn=n2, n>0, and corresponding eigenvectors sn(t)=2πsinnt.
    2. Show that the functions sn(t)=2πsinnt, n>0 form an orthonormal basis of L2([0,π]).

Notes and References

These problems were distilled from the Lecture Notes of J.H. Rubinstein on Metric and Hilbert spaces.

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