Metric and Hilbert Space Problems 2014

Last update: 4 March 2014

Homework

1. Define the standard metric on $ℂ$ and show that $ℂ,$ with this metric, is a metric space.
2. Let $d$ be the standard metric on $ℂ\text{.}$ Show that $ℝ$ is a metric subspace of $\left(ℂ,d\right)\text{.}$
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 $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{n},{d}_{n}\right)$ be metric spaces. Define the product metric $d$ on ${X}_{1}×\cdots ×{X}_{n}$ and show that $\left({X}_{1}×\cdots ×{X}_{n},d\right)$ is a metric space.
5. Let $\left(X,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ 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 ${ℝ}^{n}$ and show that ${ℝ}^{n},$ with this metric, is a metric space.
7. Define the standard norm on ${ℝ}^{n}$ and show that ${ℝ}^{n},$ with this norm, is a normed vector space.
8. Define the norm ${‖\phantom{\rule{0.5em}{0ex}}‖}_{p}$ on ${ℝ}^{n}$ and show that $\left({ℝ}^{n},{‖\phantom{\rule{0.5em}{0ex}}‖}_{p}\right)$ is a normed vector space.
9. Let $X$ be a nonempty set. Define the set of bounded functions $B\left(X,ℝ\right)$ and the sup norm on $B\left(X,ℝ\right)\text{.}$ Show that $B\left(X,ℝ\right),$ with this norm, is a normed vector space.
10. Let $a,b\in ℝ$ with $a Define the set of continuous functions $C\left(\left[a,b\right],ℝ\right)$ and the ${L}^{1}\text{-norm}$ on $C\left(\left[a,b\right],ℝ\right)\text{.}$ Show that $C\left(\left[a,b\right],ℝ\right),$ with this norm, is a normed vector space.
11. Let $a,b\in ℝ$ with $a Show that the set ${C}_{\text{bd}}\left(\left[a,b\right],ℝ\right)$ of bounded continuous functions is a metric subspace of $C\left(\left[a,b\right],ℝ\right)$ with the ${L}^{1}\text{-norm.}$
12. Let $X$ be a metric space and let ${x}_{1},{x}_{2},\dots$ be a sequence in $X\text{.}$ Show that $\underset{n\to \infty }{\text{lim}}{x}_{n}$ is unique, if it exists.
13. Let $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{\ell },{d}_{\ell }\right)$ be metric spaces. Show that a sequence $\stackrel{‾}{{x}_{n}}=\left({x}_{n}^{\left(1\right)},\dots ,{x}_{n}^{\left(\ell \right)}\right)$ in ${X}_{1}×\cdots ×{X}_{\ell }$ converges if and only if each of the sequences ${x}_{n}^{\left(i\right)}$ (in ${X}_{i}\text{)}$ converges.
14. Let $\left(X,d\right)$ be a metric space. Show that the metric $d\prime :X×X\to ℝ$ given by $d′(x,y)= d(x,y) 1+d(x,y)$ is equivalent to $d\text{.}$
15. Let $\left(X,d\right)$ be a metric space. Show that $\left(X,d\prime \right)$ 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 $d\prime$ on $X$ such that $d$ is equivalent to $d\prime$ and $\left(X,d\right)$ is not bounded and $\left(X,d\prime \right)$ is bounded.
17. Let $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{\ell },{d}_{\ell }\right)$ be metric spaces and let $\left({X}_{1}×\cdots ×{X}_{\ell },d\right)$ be the product metric space. Let $\sigma :\left({X}_{1}×\cdots ×{X}_{\ell }\right)×\left({X}_{1}×\cdots ×{X}_{\ell }\right)\to ℝ$ given by $σ(x,y)=max { di(xi,yi) | 1≤i≤ℓ } .$ Show that $\sigma$ is a metric on ${X}_{1}×\cdots ×{X}_{\ell }$ and $d$ is equivalent to $\sigma \text{.}$
18. Let $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{\ell },{d}_{\ell }\right)$ be metric spaces and let $\left({X}_{1}×\cdots ×{X}_{\ell },d\right)$ be the product metric space. Let $\rho :\left({X}_{1}×\cdots ×{X}_{\ell }\right)×\left({X}_{1}×\cdots ×{X}_{\ell }\right)\to ℝ$ be given by $ρ(x,y)= ( ∑i=1ℓdi (xi,yi)2 ) 12 .$ Show that $\rho$ is a metric on ${X}_{1}×\cdots ×{X}_{\ell }$ and $d$ is equivalent to $\rho \text{.}$
19. Let $X$ be a set and let $d$ and $d\prime$ be metrics on $X\text{.}$ Show that $d$ and $d\prime$ are equivalent if and only if $d$ and $d\prime$ satisfy the condition if $x,y\in X$ then there exist $k,k\prime \in ℝ$ such that $d(x,y)≤kd′(x,y) ≤k′d(x,y).$
20. Let $\left(X,d\right)$ be a metric space. Define the metric space topology on $X$ and show that it is a topology on $X\text{.}$
21. Let $X$ be a set and let $d$ be the discrete metric on $X\text{.}$ Determine which subsets of $X$ are in the metric space topology on $X\text{.}$
22. Give two metrics $d$ and $d\prime$ on $ℝ$ such that $ℚ$ is open in the metric space topology on $\left(ℝ,d\right)$ and $ℚ$ is not open in the metric space topology on $\left(ℝ,d\prime \right)\text{.}$
23. Let $X$ be a topological space and let $E$ be a subset of $X\text{.}$ Let ${E}^{\circ }$ be the interior of $E\text{.}$ Show that $E$ is open if and only if $E={E}^{\circ }\text{.}$
24. Let $X$ be a topological space and let $E$ be a subset of $X\text{.}$ Let ${E}^{\circ }$ be the interior of $E\text{.}$ Show that ${E}^{\circ }$ is the set of interior points of $E\text{.}$
25. Let $X$ be a topological space and let $x\in X\text{.}$ Consider the following definitions of "neighborhood of $x\text{":}$
1. A neighborhood of $x$ is a set $N\subseteq X$ such that $x\in {N}^{\circ }\text{.}$
2. A neighborhood of $x$ is a set $V\subseteq X$ such that there exists an open set $U$ of $X$ with $x\in U\subseteq V\text{.}$
Show that these two definitions of "neighborhood of $x\text{"}$ are equivalent.
26. Let $X$ be a topological space and let $E$ be a subset of $X\text{.}$ Let $\stackrel{‾}{E}$ be the closure of $E\text{.}$ Show that $E$ is closed if and only if $E=\stackrel{‾}{E}\text{.}$
27. Let $X$ be a topological space and let $E$ be a subset of $X\text{.}$ Let $x\in X\text{.}$ Show that $x$ is a closed point of $E$ if and only if there exists a sequence ${x}_{1},{x}_{2},\dots$ of points in $E$ such that $\underset{n\to \infty }{\text{lim}}{x}_{n}=x\text{.}$
28. Let $\left(X,d\right)$ be a metric space and let $x\in X$ and $r\in {ℝ}_{\text{>0}}\text{.}$ Show that the closed ball $B‾(x,r)= {y∈X | d(x,y)≤r}$ is a closed set in the metric space topology on $X\text{.}$
29. Give an example of a metric space $\left(X,d\right)$ and a point $x\in X$ such that $\stackrel{‾}{B}\left(x,1\right)\ne \stackrel{‾}{B\left(x,1\right)}\text{.}$
30. Let $\left(X,d\right)$ be a metric space and let $x\in X$ and $r\in {ℝ}_{>0}\text{.}$ Show that $\stackrel{‾}{B\left(x,r\right)}\subseteq \stackrel{‾}{B}\left(x,r\right)\text{.}$
31. In $ℝ$ with the usual topology give an example of
1. a set $A\subseteq ℝ$ which is both open and closed,
2. a set $B\subseteq ℝ$ which is open and not closed,
3. a set $C\subseteq ℝ$ which is closed and not open,
4. a set $D\subseteq ℝ$ which is not open and not closed.
32. Let $X=ℝ$ with the usual topology. Show that
1. $\left[0,1\right)\subseteq ℝ$ is not open and not closed,
2. $ℚ\subseteq ℝ$ is not open and not closed.
33. Let $X$ be a set with the discrete metric $d\text{.}$ Show that every subset of $X$ is both open and closed (in the metric space topology on $X\text{).}$
34. Let $X$ be a set and let $𝒞$ be a collection of subsets of $X\text{.}$ 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. $\varnothing \in 𝒞$ and $X\in 𝒞\text{.}$
35. Let $X$ be a topological space and let $Y\subseteq X$ with the subspace topology. Show that
1. $B\subseteq Y$ is open in $Y$ if and only if $B=Y\cap A$ for some set $A\subseteq X$ which is open in $X\text{.}$
2. $B\subseteq Y$ is closed in $Y$ if and only if there exists $F\subseteq X$ closed in $X$ such that $B=Y\cap F\text{.}$
36. Let ${X}_{1},\dots ,{X}_{\ell }$ be topological spaces and let ${X}_{1}×\cdots ×{X}_{\ell }$ have the product topology. Show that
1. If ${A}_{1}\subseteq {X}_{1},\dots ,{A}_{\ell }\subseteq {X}_{\ell }$ are open then ${A}_{1}×\cdots ×{A}_{\ell }\subseteq {X}_{1}×\cdots ×{X}_{\ell }$ is open.
2. If ${F}_{1}\subseteq {X}_{1},\dots ,{F}_{\ell }\subseteq {X}_{\ell }$ are closed then ${F}_{1}×\cdots ×{F}_{\ell }\subseteq {X}_{1}×\cdots ×{X}_{\ell }$ is closed.
37. Let $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{\ell },{d}_{\ell }\right)$ be metric spaces and let $d$ be the product metric on ${X}_{1}×\cdots ×{X}_{\ell }\text{.}$ Show that the metric space topology on $\left({X}_{1}×\cdots ×{X}_{\ell },d\right)$ is the product topology for ${X}_{1}×\cdots ×{X}_{\ell },$ where ${X}_{1},\dots ,{X}_{\ell }$ have the metric space topology.
38. Let $X$ be a topological space and let $A\subseteq X\text{.}$ Show that if $x\in X$ satisfies if $r\in {ℝ}_{>0}$ then $B\left(x,r\right)\cap A\ne \varnothing$ and $B\left(x,r\right)\cap {A}^{c}\ne \varnothing$ then $x\in \partial A\text{.}$
39. Let $X$ be a topological space and let $A\subseteq X\text{.}$ Show that $\partial A$ is a closed subset of $X\text{.}$
40. Let $X=ℝ$ with the usual topology.
1. Determine (with proof) $\partial \left(\left[0,1\right]\right)\text{.}$
2. Determine $\partial ℚ$ (with proof, of course).
41. Let $\left(X,d\right)$ be a metric space. Let $x\in X\text{.}$ Show that $\left\{x\right\}\subseteq X$ is closed (in the metric space topology on $X\text{).}$
42. Let $\left(X,d\right)$ be a metric space and let $x\in X\text{.}$ Show that $x$ is isolated if and only if there exists $\epsilon \in {ℝ}_{>0}$ such that $B\left(x,\epsilon \right)=\left\{x\right\}\text{.}$
43. Let $X=ℝ$ with the usual topology. Show that
1. ${ℤ}_{>0}\subseteq ℝ$ is a discrete set in $ℝ,$
2. $\left\{\frac{1}{n} | n\in {ℤ}_{>0}\right\}\subseteq ℝ$ is a discrete set in $ℝ\text{.}$
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 $X=ℝ$ with the usual topology.
1. Show that $ℚ$ is dense in $ℝ\text{.}$
2. Show that ${ℚ}^{c}$ is dense in $ℝ\text{.}$
3. Show that ${ℤ}_{>0}$ is nowhere dense in $ℝ\text{.}$
4. Show that $ℤ$ is nowhere dense in $ℝ\text{.}$
5. Show that $ℝ$ is nowhere dense in ${ℝ}^{2}\text{.}$
47. Let $C$ be the Cantor set in $ℝ,$ where $ℝ$ has the usual topology.
1. Show that $C$ is closed in $ℝ\text{.}$
2. Show that $C$ does not contain any interval in $ℝ\text{.}$
3. Show that $C$ has nonempty interior.
4. Show that $C$ is nowhere dense in $ℝ\text{.}$
48. Let $\left(X,d\right)$ and $\left(Y,\rho \right)$ be metric spaces and let $f:X\to Y$ be a function. Let $a\in X\text{.}$ Show that $f$ is continuous at $a$ if and only if $f$ satisfies: if $\epsilon \in {ℝ}_{>0}$ then there exists $\delta \in {ℝ}_{>0}$ such that
if $x\in X$ and $d\left(x,a\right)<\delta$ then $\rho \left(f\left(x\right),f\left(a\right)\right)<\epsilon \text{.}$
49. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a function. Show that $f$ is continuous if and only if $f$ satisfies: if $a\in X$ then $f$ is continuous at $a\text{.}$
50. Let $X$ and $Y$ be metric spaces and let $f:X\to Y$ be a function. Let $a\in X\text{.}$ Show that $f$ is continuous at $a$ if and only if $f$ satisfies: if $\epsilon \in {ℝ}_{>0}$ then there exists $\delta \in {ℝ}_{>0}$ such that $f(B(a,δ))⊆ B(f(a),ε).$
51. Let $X$ and $Y$ be metric spaces and let $f:X\to Y$ be a function. Let $a\in X\text{.}$ Show that $f$ is continuous at $a$ if and only if $f$ satisfies if ${x}_{1},{x}_{2},\dots$ is a sequence in $X$ and $\underset{n\to \infty }{\text{lim}}{x}_{n}={x}_{0}$ then $\underset{n\to \infty }{\text{lim}}f\left({x}_{n}\right)=f\left({x}_{0}\right)\text{.}$
52. Let $X$ and $Y$ be metric spaces and let $f:X\to Y$ be a function. Let $a\in X\text{.}$ Show that $f$ is continuous at $a$ if and only if $f$ satisfies: if ${x}_{1},{x}_{2},\dots$ is a convergent sequence in $X$ then $\underset{n\to \infty }{\text{lim}}f\left({x}_{n}\right)=f\left(\underset{n\to \infty }{\text{lim}}{x}_{n}\right)\text{.}$
53. Let $X$ and $Y$ be topological spaces. Let $f:X\to Y$ be a function. Show that $f$ is continuous if and only if $f$ satisfies: if $F\subseteq Y$ is closed then ${f}^{-1}\left(F\right)$ is closed in $X\text{.}$
54. Let $X,Y$ and $Z$ be topological spaces and let $f:X\to Y$ and $g:Y\to Z$ be continuous functions. Show that $g\circ f$ is a continuous function.
55. Let $X,Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Let $A\subseteq X\text{.}$ Show that the restriction of $f$ to $A,$ $f{|}_{A}:A\to Y$ is continuous.
56. Let $\left(X,d\right),\left({Y}_{1},{\rho }_{1}\right)$ and $\left({Y}_{2},{\rho }_{2}\right)$ be metric spaces. Let $f:X\to {Y}_{1}$ and $g:X\to {Y}_{2}$ be functions. Define $h:X\to {Y}_{1}×{Y}_{2}$ by $h\left(x\right)=\left(f\left(x\right),g\left(x\right)\right)\text{.}$ Let $a\in X\text{.}$ Show that $h$ is continuous at $a$ if and only if $f$ and $g$ are continuous at $a\text{.}$ $X ⟶Δ X×X ↓f×g Y1×Y2 x ⟼ (x,x) ↧ (f(x),g(x))$
57. Let $\left(X,d\right),\left({Y}_{1},{\rho }_{1}\right)$ and $\left({Y}_{2},{\rho }_{2}\right)$ be metric spaces. Let $f:X\to {Y}_{1}$ and $g:X\to {Y}_{2}$ be functions. Define $h:X\to {Y}_{1}×{Y}_{2}$ by $h\left(x\right)=\left(f\left(x\right),g\left(x\right)\right)\text{.}$ Let $a\in X\text{.}$ Show that $h$ is continuous if and only if $f$ and $g$ are continuous.
58. Let $\left(X,d\right)$ be a metric space and let $f:X\to ℝ$ and $g:X\to ℝ$ 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 $x\in X$ then $g\left(x\right)\ne 0$ then $f/g$ is continuous.
59. Let $X$ be a topological space and let $f:X\to ℝ$ and $g:X\to ℝ$ 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 $x\in X$ then $g\left(x\right)\ne 0$ then $f/g$ is continuous.
60. Let $\left(X,d\right)$ be a metric space. Show that $d:X×X\to ℝ$ is continuous.
61. Let $f:ℝ×ℝ\to ℝ$ be given by $f(x,y)= { xyx2+y2, if (x,y)≠ (0,0) 0, if (x,y)= (0,0).$ If $a\in ℝ$ let ${\ell }_{a}:ℝ\to ℝ$ be given by ${\ell }_{a}\left(y\right)=f\left(a,y\right)\text{.}$ If $b\in ℝ$ let ${r}_{b}:ℝ\to ℝ$ be given by ${r}_{b}\left(x\right)=f\left(x,b\right)\text{.}$
1. Let $a\in ℝ\text{.}$ Show that ${\ell }_{a}:ℝ\to ℝ$ is continuous.
2. Let $b\in ℝ\text{.}$ Show that ${r}_{b}:ℝ\to ℝ$ is continuous.
3. Show that $f$ is not continuous at $\left(0,0\right)\text{.}$
62. Give an example of metric spaces $X,Y$ and $Z$ and a function $f:X×Y\to ℤ$ such that
1. if $x\in X$ then $ℓx: Y ⟶ Z y ⟼ f(x,y)$ is continuous,
2. if $y\in Y$ then $ry: X ⟶ Z x ⟼ f(x,y)$ is continuous,
3. $f:X×Y\to Z$ is not continuous.
63. Let $X$ be a topological space and let $A\subseteq X$ and $B\subseteq X$ be closed subsets of $X$ such that $X=A\cup B\text{.}$ Let $Y$ be a topological space and let $f:A\to Y$ and $g:B\to Y$ be continuous functions such that if $x\in A\cap B$ then $f\left(x\right)=g\left(x\right)\text{.}$ Define $h:X\to Y$ by $h(x)= { f(x), if x∈A, g(x), if x∈B.$ Show that $h:X\to Y$ is continuous.
64. Show that the function $f:ℝ\to ℝ$ given by $f(x)= x1+x2,$ is uniformly continuous.
65. Show that the function $f:ℝ\to ℝ$ given by $f\left(x\right)={x}^{2},$ is not uniformly continuous.
66. Let $\left(X,d\right)$ and $\left(Y,\rho \right)$ be metric spaces and let $f:X\to Y$ be a function. Show that if $f$ is uniformly continuous then $f$ is continous.
67. Let $\left(X,d\right)$ and $\left(Y,\rho \right)$ be metric spaces. Let $\left\{{f}_{k}\right\}$ be a sequence of functions ${f}_{k}:X\to Y$ and let $f:X\to Y$ be a function. Show that $\left\{{f}_{k}\right\}$ converges uniformly to $f$ if and only if $\text{sup}\left\{\rho \left({f}_{k}\left(x\right),f\left(x\right)\right) | x\in X\right\}\to 0\text{.}$
68. Let $\left\{{f}_{k}\right\}$ be a sequence of continuous functions from a metric space $\left(X,d\right)$ to a metric space $\left(Y,\rho \right)\text{.}$ Suppose that $\left\{{f}_{k}\right\}$ converges uniformly to $f:X\to Y\text{.}$ Show that $f:X\to Y$ is continuous.
69. Let $\left(X,d\right)$ be a metric space and let ${x}_{1},{x}_{2},\dots$ be a sequence in $X\text{.}$ Show that if $\left({x}_{1},{x}_{2},\dots \right)$ is a Cauchy sequence then $\left\{{x}_{1},{x}_{2},\dots \right\}$ is bounded.
70. Let $\left(X,d\right)$ be a metric space and let $\left({x}_{1},{x}_{2},\dots \right)$ be a sequence in $X\text{.}$ Show that if $\left({x}_{1},{x}_{2},\dots \right)$ converges then $\left({x}_{1},{x}_{2},\dots \right)$ is a Cauchy sequence.
71. Let $\left(X,d\right)$ be a metric space and let $\left({x}_{1},{x}_{2},\dots \right)$ be a sequence in $X\text{.}$ Show that if $\left({x}_{1},{x}_{2},\dots \right)$ is a Cauchy sequence and contains a convergent subsequence then $\left({x}_{1},{x}_{2},\dots \right)$ converges.
72. Give an example of a metric space $\left(X,d\right)$ and a Cauchy sequence $\left({x}_{1},{x}_{2},\dots \right)$ in $X$ that does not converge.
73. Give an example of a metric space $\left(X,d\right)$ that is not complete.
74. Show that $ℝ$ with the usual metric is a complete metric space.
75. Let $\left(X,d\right)$ be a complete metric space. Let $Y\subseteq X$ be a subspace of $X\text{.}$ Show that if $Y$ is closed then $\left(Y,d\right)$ is complete.
76. Give an example of a metric space $\left(X,d\right)$ and a subspace $Y\subseteq X$ such that $\left(X,d\right)$ is a complete metric space and $\left(Y,d\right)$ is not complete.
77. Let $\left(X,d\right)$ be a metric space and let $Y\subseteq X$ be a subspace of $X\text{.}$ Show that if $\left(Y,d\right)$ is complete then $Y$ is a closed subset of $X\text{.}$
78. Let $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{\ell },{d}_{\ell }\right)$ be metric spaces and let $\left({X}_{1}×\cdots ×{X}_{\ell },d\right)$ be the product metric space. Show that if $\left({X}_{1},{d}_{1}\right),\dots ,\left({X}_{\ell },{d}_{\ell }\right)$ are complete then $\left({X}_{1}×\cdots ×{X}_{\ell },d\right)$ is complete.
79. Let $\left(X,d\right)$ and $\left(Y,d\prime \right)$ be metric spaces and let ${C}_{b}\left(X,Y\right)$ be the set of bounded continuous functions $f:X\to Y$ with the metric given by $ρ(f,g)=sup { d′(f(x),g(x)) | x∈X } .$ Show that if $\left(Y,d\prime \right)$ is complete then $\left({C}_{b}\left(X,y\right),\rho \right)$ is a complete metric space.
80. Let $\left(X,d\right)$ and $\left(Y,d\prime \right)$ be metric spaces and let ${C}_{b}\left(X,Y\right)$ be the set of bounded continuous functions $f:X\to Y$ with the metric $\rho :{C}_{b}\left(X,Y\right)×{C}_{b}\left(X,Y\right)\to {ℝ}_{>0}$ given by $ρ(f,g)=sup { d′(f(x),g(x)) | x∈X } .$ Show that $\left({C}_{b}\left(X,Y\right),\rho \right)$ is a metric space.
81. Let $\left(X,d\right)$ be a metric space and let $U\subseteq X$ and $V\subseteq X\text{.}$ Show that if $U$ and $V$ are open and dense then $U\cap V$ is open and dense.
82. Let $X=ℝ$ with the usual metric and let $U=ℚ$ and $V={ℚ}^{c}\text{.}$ Show that $U$ and $V$ are dense and $U\cap B=\varnothing \text{.}$
83. Let $X=ℚ$ with the usual metric and let $ℚ=\left\{{q}_{1},{q}_{2},{q}_{3},\dots \right\}$ be an enumeration of $ℚ\text{.}$ For $n\in {ℤ}_{>0}$ let ${Q}_{n}=ℚ\\left\{{q}_{n}\right\}\text{.}$
1. Show that if $n\in {ℤ}_{>0}$ then ${Q}_{n}$ is open and dense.
2. Show that $\bigcap _{n\in {ℤ}_{>0}}{Q}_{n}=\varnothing \text{.}$
84. Let $\left(X,d\right)$ be a complete metric space and let $\left\{{U}_{1},{U}_{2},{U}_{3},\dots \right\}$ be a sequence of open and dense subsets of $X\text{.}$ show that $\bigcap _{n\in {ℤ}_{>0}}{U}_{n}$ is dense in $X\text{.}$
85. Let $\left(X,d\right)$ be a complete metric space and let $\left\{{F}_{1},{F}_{2},{F}_{3},\dots \right\}$ be a sequence of nowhere dense subsets of $X\text{.}$ Show that $\bigcup _{n\in {ℤ}_{>0}}{F}_{n}$ has empty interior.
86. Show that $ℝ,$ with the standard topology, cannot be be written as a countable union of nowhere dense sets.
87. Let $X=ℚ,$ with the standard topology. Let $ℚ=\left\{{q}_{1},{q}_{2},{q}_{3},\dots \right\}$ be an enumeration of $ℚ\text{.}$
1. Show that $\left\{{q}_{n}\right\}$ is nowhere dense.
2. Determine the interior of $\bigcup _{n\in {ℤ}_{>0}}\left\{{q}_{n}\right\}\text{.}$
88. Let $\left(X,d\right)$ be a complete metric space and let $\left\{{f}_{1},{f}_{2},{f}_{3},\dots \right\}$ be a sequence of continuous functions $fn:X→ℝ,for n∈ℤ>0.$ Assume that if $x\in X$ then $\left\{{f}_{1}\left(x\right),{f}_{2}\left(x\right),\dots \right\}$ is bounded in $X\text{.}$ Show that there exists and open set $U\subseteq X$ such that there exists $M\in {ℝ}_{>0}$ such that
if $x\in U$ and $n\in {ℤ}_{>0}$ then $|{f}_{n}\left(x\right)|\le M\text{.}$
89. Show that the completion of $\left(0,1\right)$ with the usual metric is $\left[0,1\right]$ with the usual metric.
90. Let $\left(X,d\right)$ and $\left(Y,\rho \right)$ be metric spaces and let $f:X\to Y$ be an isometry. Show that $f$ is injective.
91. Give an example of an isometry $f:X\to Y$ that is not surjective.
92. Let $\left(X,d\right)$ be a metric space. Show that a completion of $\left(X,d\right)$ exists.
93. Let $\left(X,d\right)$ be a metric space. Show that the completion of $\left(X,d\right)$ is unique (if it exists).
94. Let $\left(X,d\right)$ be a metric space. Let $\left(\left({X}_{1},{d}_{1}\right),{\phi }_{1}\right)$ and $\left(\left({X}_{2},{d}_{2}\right),{\phi }_{2}\right)$ be completions of $\left(X,d\right)\text{.}$ Show that there is a surjective isometry $f:{X}_{1}\to {X}_{2}$ such that $f\circ {\phi }_{1}={\phi }_{2}\text{.}$
95. Let $X=\left\{0,1\right\}$ and let $𝒯=\left\{\varnothing ,X,\left\{0\right\}\right\}\text{.}$
1. Show $𝒯$ is a topology on $X\text{.}$
2. Show that there does not exist a metric $d:X×X\to {ℝ}_{\ge 0}$ such that $𝒯$ is the metric space topology of $\left(X,d\right)\text{.}$
96. Let $X$ be a complete metric space and let $f:X\to X$ be a contraction. Show that $f$ has a unique fixed point.
97. Let $\alpha \in ℝ$ with $0<\alpha <1\text{.}$ Let $X$ be a complete metric space and let $f:X\to X$ be a $\alpha \text{-contraction.}$ Let $x\in X,$ ${x}_{0}=x$ and ${x}_{n+1}=f\left({x}_{n}\right),$ for $n\in {ℤ}_{\ge 0}\text{.}$
1. Show that the sequence ${x}_{0},{x}_{1},{x}_{2},\dots$ converges in $X\text{.}$
Let $p=\underset{n\to \infty }{\text{lim}}{x}_{n}\text{.}$
1. Show that $d\left(x,p\right)\le \frac{d\left(x,f\left(x\right)\right)}{1-\alpha }\text{.}$
2. Show that $f\left(p\right)=p\text{.}$
98. Let $U$ be an open subset of ${ℝ}^{2}\text{.}$ Let $f:U\to ℝ$ be a continuous function which satisfies the Lipschitz condition with respect to the second variable: There exists $\alpha \in {ℝ}_{>0}$ such that
if $\left(x,{y}_{1}\right),\left(x,{y}_{2}\right)\in U$ then $|f\left(x,{x}_{1}\right)-f\left(x,{y}_{2}\right)|\le \alpha |{y}_{1},{y}_{2}|\text{.}$
Show that if $\left({x}_{0},{y}_{0}\right)\in U$ then there exists $\delta \in {ℝ}_{>0}$ such that $y\prime \left(x\right)=f\left(x,y\left(x\right)\right)$ has a unique solution $y:\left[{x}_{0}-\delta ,{x}_{0}+\delta \right]\to ℝ$ such that $y\left({x}_{0}\right)={y}_{0}\text{.}$
99. Let $\left(X,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a normed vector space. Show that $\left(X,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ is complete if and only if every norm absolutely convergent series is convergent in $X\text{.}$
100. Let $S$ be the set of linear combinations of step functions $f:{ℝ}^{k}\to ℝ\text{.}$ Let $‖f‖=∫|f| andd(f,g)= ‖f-g‖$ for $f,g\in S\text{.}$
1. show that $‖\phantom{\rule{0.5em}{0ex}}‖:S\to {ℝ}_{\ge 0}$ is not a norm on $S\text{.}$
2. Show that $d:S×S\to {ℝ}_{\ge 0}$ is not a metric on $S\text{.}$
101. Let $S$ be the set of linear combinations of step functions $f:{ℝ}^{k}\to ℝ\text{.}$ Let $\sum _{i\in {ℤ}_{>0}}{f}_{i}$ be a series in $S$ which is norm absolutely convergent. Show that there exists a full set in ${ℝ}^{k}$ on which $\sum _{i\in {ℤ}_{\ge 0}}{f}_{i}$ converges.
102. Let $S$ be the set of linear combinations of step functions $f:{ℝ}^{k}\to ℝ\text{.}$ Let $\sum _{n\in {ℤ}_{>0}}{f}_{k}$ be a series in $S$ which is norm absolutely convergent. Show that $\sum _{n\in {ℤ}_{>0}}{f}_{n}=0$ almost everywhere if and only if the limit of the norms of the partial sums of ${f}_{n}$ converge to $0\text{.}$
103. Let $L\prime$ 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 $f:{ℝ}^{k}\to ℝ\text{.}$ Define $‖f‖=∫fand d(f,g)=‖f-g‖$ for $f,g\in L\prime \text{.}$
1. Show that $‖\phantom{\rule{0.5em}{0ex}}‖:L\prime \to {ℝ}_{\ge 0}$ is a norm on $L\prime \text{.}$
2. Show that $d:L\prime ×L\prime \to {ℝ}_{\ge 0}$ is a metric on $L\prime \text{.}$
104. Let $I$ be a closed and bounded interval in $ℝ\text{.}$ Let ${x}_{1},{x}_{2},{x}_{3},\dots ,$ be a sequence in $I\text{.}$ Show that there exists a subsequence ${x}_{{n}_{1}},{x}_{{n}_{2}},{x}_{{n}_{3}},\dots$ of ${x}_{1},{x}_{2},{x}_{3},\dots$ such that ${x}_{{n}_{1}},{x}_{{n}_{2}},{x}_{{n}_{3}},\dots$ converges in $I\text{.}$
105. Let $X$ be a compact topological space. Let $C$ be a closed subset of $X\text{.}$ Show that $C$ is compact.
106. Let $X$ be a metric space and let $E$ be a compact subset of $X\text{.}$ Show that $E$ is closed and bounded.
107. Let $C\left(\left[0,1\right],ℝ\right)=\left\{f:\left[0,1\right]\to ℝ | f \text{is continuous}\right\}$ and let $d(f,g)=sup { |f(x)-g(x)| | x∈[0,1] } .$
1. Show that $d:C\left(\left[0,1\right],ℝ\right)×C\left(\left[0,1\right],ℝ\right)$ is a metric on $C\left(\left[0,1\right],ℝ\right)\text{.}$
2. Let $A={\stackrel{‾}{B}}_{1}\left(0\right)=\left\{f\in C\left(\left[0,1\right],ℝ\right) | d\left(f,0\right)\le 1\right\}\text{.}$ Show that $A$ is closed and bounded.
3. Show that $A$ is not compact.
108. Let $K\subseteq ℝ\text{.}$ Show that $K$ is compact if and only if $K$ is closed and bounded.
109. Let $\left(X,d\right)$ and $\left(Y,d\prime \right)$ be metric spaces and let $f:X\to Y$ be a continuous function. Let $K$ be a compact subset of $X\text{.}$ Show that $f\left(K\right)$ is compact in $Y\text{.}$
110. Let $X$ be a compact metric space. Let $f:X\to ℝ$ be a continuous function. Show that $f$ attains a maximum and a minimum value.
111. Let $X$ be a compact metric space. Let $f:X\to Y$ 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 $A\subseteq X\text{.}$ Show that if $A$ is totally bounded then $A$ is bounded.
114. Let $X=ℝ$ 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 $A\subseteq X\text{.}$ 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 $\ell \in {ℤ}_{>0}$ and ${C}_{1},\dots ,{C}_{\ell }\in 𝒞$ then ${C}_{1}\cap \cdots \cap {C}_{\ell }\ne \varnothing$ then $\bigcap _{C\in 𝒞}C\ne \varnothing \text{.}$
117. Let $X$ be a topological space and let $K\subseteq X\text{.}$ Assume $X$ is compact. Show that if $K$ is closed then $K$ is compact.
118. Let $X$ be a topological space and let $K\subseteq X\text{.}$ 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 $f:X\to Y$ be a continuous function. Let $K\subseteq X\text{.}$ Show that if $K$ is compact then $f\left(K\right)$ is compact.
121. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Assume $f$ is a bijection, $X$ is compact and $Y$ is Hausdorff. Show that the inverse function ${f}^{-1}:Y\to X$ is continuous.
122. Let $X=\left[0,2\pi \right)$ and $Y={S}^{1}=\left\{\left(x,y\right)\in {ℝ}^{2} | {x}^{2}+{y}^{2}=1\right\}\text{.}$ Let $f:\left[0,2\pi \right)\to {S}^{1}$ be given by $f(x)=(cos x,sin x).$
1. Show that $f$ is continuous.
2. Show that $f$ is a bijection.
3. Show that ${f}^{-1}:{S}^{1}\to \left[0,2\pi \right)$ is not continuous.
4. Why does this not contradict the previous problem?
123. Let $X$ be a set with $\text{Card}\left(X\right)>1\text{.}$
1. Show that $X$ with the discrete topology is disconnected.
2. Show that $X$ with the indiscrete topology is connected.
124. Let ${X}_{1}$ and ${X}_{2}$ be the subspaces of $ℝ$ given by ${X}_{1}-ℝ\\left\{0\right\}$ and ${X}_{2}=ℚ\text{.}$ Show that ${X}_{1}$ and ${X}_{2}$ are disconnected.
125. Let $Y=\left\{0,1\right\}$ with the discrete topology. Let $X$ be a topological space. Show that $X$ is connected if and only if every continuous function $f:X\to Y$ is constant.
126. Let $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous function. Let $E\subseteq X\text{.}$ Show that if $E$ is connected then $f\left(E\right)$ is connected.
127. Let $X$ be a connected topological space and let $A\subseteq X\text{.}$ Show that if $A$ is connected then the closure of $A,$ $\stackrel{‾}{A},$ is connected.
128. Let $A=\left(-\infty ,0\right)$ and $B=\left(0,\infty \right)$ as subsets of $ℝ\text{.}$ Show that $A$ is connected, $B$ is connected and $A\cup B$ is not connected.
129. Let $X$ be a topological space. Let $𝒮$ be a collection of subsets of $X$ such that $\bigcap _{A\in 𝒮}A\ne \varnothing \text{.}$ Show that $\bigcup _{A\in 𝒮}A$ is connected.
130. Let $X$ be a topological space such that if $x,y\in X$ then there exists $A\subseteq X$ such that $x\in A,$ $y\in A$ and $A$ is connected. Show that $X$ is connected.
131. Let $X$ be a topological space. For $x\in X$ let ${C}_{x}$ be the connected component containing $x\text{.}$
1. Let $y\in X\text{.}$ Show that ${C}_{y}$ is connected and closed.
2. Show that the connected components of $X$ partition $X\text{.}$
132. Let $X$ be a set with the discrete topology. Determine (with proof) the connected components of $X$.
133. Show that a subset of $ℝ$ 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 $f:X\to ℝ$ be a continuous function. Show that if $x,y\in X$ and $r\in ℝ$ such that $f\left(x\right)\le r\le f\left(y\right)$ then there exists $c\in X$ such that $f\left(c\right)=r\text{.}$
137. Let $X$ be a topological space. Show that if $X$ is path connected then $X$ is connected.
138. Let $X=\left\{\left(t,\text{sin}\left(\frac{\pi }{t}\right)\right) | t\in \left(0,2\right]\right\}\text{.}$
1. Let $\phi :{ℝ}^{2}\to ℝ$ be given by $\phi \left(x,y\right)=x\text{.}$ Show that $\phi :X\to \left(0,2\right]$ is a homeomorphism.
2. Show that $X$ is connected.
3. Show that $\stackrel{‾}{X}$ is connected.
4. Show that $\stackrel{‾}{X}$ is not path connected.
139. Let $p\in {ℝ}_{\ge 1}\text{.}$ Let $V={ℝ}^{n}$ with $‖(a1,…,an)‖p= ( |a1|p +⋯+ |an|p ) 1p .$ Show that $\left(V,{‖\phantom{\rule{0.5em}{0ex}}‖}_{p}\right)$ is a Banach space.
140. Let $V={ℝ}^{n}$ with $‖(a1,…,an)‖∞= sup{|a1|,|a2|,…,|an|}.$ Show that $\left(V,{‖\phantom{\rule{0.5em}{0ex}}‖}_{\infty }\right)$ is a Banach space.
141. Let ${\ell }^{p}$ be the vector space of sequences $\left({a}_{1},{a}_{2},\dots \right)$ in $ℝ$ such that $\sum _{n\in {ℤ}_{>0}}{|{a}_{n}|}^{p}<\infty \text{.}$ Let $‖(a1,a2,…)‖q= (∑n∈ℤ>0|an|p)1p.$ Show that $\left({\ell }^{p},{‖\phantom{\rule{0.5em}{0ex}}‖}_{p}\right)$ is a Banach space.
142. Let ${\ell }^{\infty }$ be the vector space of bounded sequences $\left({a}_{1},{a}_{2},\dots \right)$ in $ℝ$ with $‖(a1,a2,…)‖∞=sup {|ai| | i∈ℤ>0}.$ Show that $\left({\ell }^{\infty },{‖\phantom{\rule{0.5em}{0ex}}‖}_{\infty }\right)$ is a Banach space.
143. Let $X$ be a topological space. Let $F=ℝ$ or $ℂ\text{.}$ Let ${C}_{b}\left(X,F\right)$ be the vector space of bounded continuous functions with $‖f‖=sup {|f(x)| | x∈X}.$ Show that $\left({C}_{b}\left(X,F\right),‖\phantom{\rule{0.5em}{0ex}}‖\right)$ is a Banach space.
144. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a normed vector space. Show that $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ is a Banach space if and only if every norm absolutely convergent series is convergent.
145. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a normed vector space. Show that a Schauder basis of $V$ is a total set.
146. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a normed vector space. Show that if $V$ has a Schauder basis then $V$ is separable.
147. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ 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 $V\text{.}$
148. Let ${e}_{i}=\left(0,0,\dots ,0,1,0,0,\dots \right)$ with $1$ in the ${i}^{\text{th}}$ entry. Show that $\left\{{e}_{1},{e}_{2},{e}_{3},\dots \right\}$ is a Schauder basis of ${\ell }^{p}\text{.}$
149. Show that ${\ell }^{\infty }$ is not separable.
150. Show that ${\ell }^{\infty }$ does not have a Schauder basis.
151. Let ${C}_{b}\left(\left[0,1\right],ℝ\right)$ be the vector space of bounded continuous functions on $\left[0,1\right]\text{.}$
1. Show that the set of polynomials is dense in the space of continuous functions on $\left[0,1\right]$ with the supremum norm.
2. Show that the polynomials with rational coefficients form a countable dense set in ${C}_{b}\left(\left[0,1\right],ℝ\right)\text{.}$
3. Show that ${C}_{b}\left(\left[0,1\right],ℝ\right)$ is separable.
152. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a Banach space.
1. Show that if $\text{dim}\left(V\right)<\infty$ 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 ${‖\phantom{\rule{0.5em}{0ex}}‖}_{1}$ and ${‖\phantom{\rule{0.5em}{0ex}}‖}_{2}$ be norms on $V\text{.}$ Show that ${‖\phantom{\rule{0.5em}{0ex}}‖}_{1}$ and ${‖\phantom{\rule{0.5em}{0ex}}‖}_{2}$ are equivalent.
154. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a normed vector space. Show that if $V$ is finite dimensional then the closed unit ball in $V$ is compact.
155. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be an infinite dimensional Banach space. Construct a sequence $\left({e}_{1},{e}_{2},\dots \right)$ of unit vectors in $V$ such that if $i,j\in {ℤ}_{>0}$ and $i\ne j$ then $d\left({e}_{i},{e}_{j}\right)>\frac{1}{2}\text{.}$
156. Let $V={ℂ}^{n}$ with $⟨ (x1,x2,…,xn), (y1,y2,…,yn) ⟩ =∑i=1nxi yi‾.$ show that $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ is a Hilbert space.
157. Let ${\ell }^{2}$ be the set of sequences $\left({a}_{1},{a}_{2},\dots \right)$ in $ℂ$ such that $\sum _{i\in {ℤ}_{>0}}{|{a}_{i}|}^{2}<\infty \text{.}$ Let $⟨ (x1,x2,…), (y1,y2,…) ⟩ =∑i∈ℤ>0 xiyi‾.$ Show that $\left({\ell }^{2},⟨\phantom{\rule{1em}{0ex}}⟩\right)$ is a Hilbert space.
158. Carefully define the space ${L}^{2}\left(\left[a,b\right]\right)$ and show that if $⟨f,g⟩= ∫abf(t) g(t)‾dt$ then $\left({L}^{2}\left(\left[a,b\right]\right),⟨,⟩\right)$ is a Hilbert space.
159. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be an inner product space and let $‖x‖=⟨x,x⟩$ for $x\in V\text{.}$ Show that if $x,y\in V$ then ${‖x+y‖}^{2}+{‖x-y‖}^{2}=2{‖x‖}^{2}+2{‖y‖}^{2}\text{.}$
160. Let $\left(V,‖\phantom{\rule{0.5em}{0ex}}‖\right)$ be a normed vector space. Define $⟨x,y⟩=14 ( ‖x+y‖2- ‖x-y‖2+i ‖x+iy‖2-i ‖x-iy‖2 )$ for $x,y\in V\text{.}$ Show that if $‖\phantom{\rule{0.5em}{0ex}}‖$ satisfies if $x,y\in V$ then ${‖x+y‖}^{2}+{‖x-y‖}^{2}=2{‖x‖}^{2}+2{‖y‖}^{2}$ then $⟨\phantom{\rule{1em}{0ex}}⟩$ is a norm on $V\text{.}$
161. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be an inner product space. Show that the Gram-Schmidt process produces an orthonormal basis of $V\text{.}$
162. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be an inner product space. Let $W$ be a vector subspace of $V\text{.}$ Show that if $W$ admits an orthogonal projection $P$ then $P$ is unique.
163. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a Hilbert space. Let $W$ be a vector subspace of $V\text{.}$ Show that if there is an orthogonal projection $P$ onto $W$ then $W$ is closed.
164. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a Hilbert space. Let $\left({a}_{1},{a}_{2},\dots \right)$ be an orthonormal sequence in $V\text{.}$ Let $W=\text{span}\left\{{a}_{1},{a}_{2},\dots \right\}$ and let $M=\stackrel{‾}{W}$ be the closure of $W\text{.}$ Show that $P:V\to V$ given by $P(x)=∑n∈ℤ>0 ⟨x,an⟩an$ is an orthonormal projection onto $M\text{.}$
165. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a Hilbert space. Let $\left({a}_{1},{a}_{2},\dots \right)$ be an orthonormal sequence in $V\text{.}$ Let $x\in V\text{.}$ Show that $P(x)=∑n∈ℤ>0 ⟨x,an⟩an$ is independent of the order of the terms in the sum.
166. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a Hilbert space. Let $\left({a}_{1},{a}_{2},\dots \right)$ be an orthonormal sequence in $V\text{.}$ Let $x\in V\text{.}$ Show that $∑n∈ℤ>0 |⟨x,an⟩|2 ≤‖x‖2.$
167. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a separable Hilbert space. Show that $V$ has a Schauder basis.
168. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a Hilbert space. Assume that $V$ has a countable orthonormal set $\left\{{a}_{1},{a}_{2},\dots \right\}$ which is a total set. Show that $\left\{{a}_{1},{a}_{2},\dots \right\}$ is a Schauder basis for $V\text{.}$
169. Show that the functions $em(t)= 12πeimt, for m∈ℤ,$ form an orthonormal basis of ${L}^{2}\left(\left[0,2\pi \right]\right)\text{.}$
170. Let $\left(V,⟨\phantom{\rule{1em}{0ex}}⟩\right)$ be a Hilbert space and let $W$ be a closed subspace of $V\text{.}$ Show that $V=W\oplus {W}^{\perp }\text{.}$
171. Let $V$ and $W$ be normed vector spaces and let $T:V\to W$ be a linear transformation. Show that if $V$ is finite dimensional then $T$ is bounded.
172. Let $V=C\left(\left[0,1\right]\right)$ be the vector space of continuous functions $f:\left[0,1\right]\to ℝ$ with norm given by $‖f‖=∫01 |f(t)|dt.$ Let $T:V\to ℝ$ be given by $T\left(f\right)=f\left(0\right)\text{.}$
1. Show that $V$ is infinite dimensional.
2. Show that $T$ is not bounded.
173. Let $V,W$ be normed vector spaces and let $T:V\to W$ be a linear transformation. Show that if $T$ is continuous then $T$ is bounded.
174. Let $V,W$ be normed vector spaces and let $T:V\to W$ 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 ${\text{id}}_{V}:V\to V$ has operator norm $1$ and the zero operator $0:V\to W$ has operator norm $0\text{.}$
176. Let ${\ell }^{\infty }$ be the vector space of bounded sequences $\left({a}_{1},{a}_{2},\dots \right)$ in $ℝ$ with norm given by $‖(a1,a2,…)‖=sup {|a1|,|a2|,…}.$ Let $\left({\lambda }_{1},{\lambda }_{2},\dots \right)$ be a bounded sequence in $ℝ\text{.}$ Define $T:{\ell }^{\infty }\to {\ell }^{\infty }$ 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 $a,b\in ℝ$ with $a Let $k:\left[a,b\right]×\left[a,b\right]\to ℂ$ be a continuous function. Let $X=\left\{f:\left[a,b\right]\to ℂ | f \text{is continuous}\right\}$ with the supremum norm. Define $T:X\to X$ by $\left(Tf\right)\left(t\right)={\int }_{a}^{b}k\left(t,s\right)x\left(s\right)ds\text{.}$
1. Show that $X$ is a Banach space.
2. Show that if $f\in X$ then $Tf\in X\text{.}$
3. Show that $T$ is a bounded linear transformation.
178. Let $V,W$ be normed vector spaces. Let $B\left(V,W\right)$ be the vector space of bounded linear operators with the operator norm. Show that if $W$ is a Banach space then $B\left(V,W\right)$ is a Banach space.
179. Let $a,b\in ℝ$ with $a Let $C\left(\left[a,b\right]\right)$ be the vector space of continuous functions $f:\left[a,b\right]\to ℂ$ with the supremum norm. Define $T:C\left(\left[a,b\right]\right)\to ℂ$ by $Tf=∫abf(t)dt$ Show that the operator norm of $T$ is $‖T‖=b-a.$
180. Let ${H}_{1}$ and ${H}_{2}$ be Hilbert spaces and let $T:{H}_{1}\to {H}_{2}$ be a bounded linear transformation.
1. Show that there exists a unique function ${T}^{*}:{H}_{2}\to {H}_{1}$ such if $x\in {H}_{1}$ and $y\in {H}_{2}$ then $⟨Tx,y⟩2= ⟨x,T*y⟩1.$
2. Show that ${T}^{*}$ is a linear transformation.
3. Show that ${T}^{*}$ is bounded.
4. Show that $‖{T}^{*}‖=‖T‖\text{.}$
181. Let $H$ be a Hilbert space and let $f:H\to ℂ$ be a bounded linear functional. Show that there exists a unique $a\in H$ such that if $x\in H$ then $f\left(x\right)=⟨x,a⟩\text{.}$
182. Let ${H}_{1}$ and ${H}_{2}$ be Hilbert spaces and let $T:{H}_{1}\to {H}_{2}$ be a bounded linear transformation. Show that $T**=T.$
183. Let $a,b\in ℝ$ with $a Let $C\left(\left[a,b\right]\right)$ be the Banach space of continuous functions $f:\left[a,b\right]\to ℝ$ with the supremum norm. Let ${t}_{0}\in \left[a,b\right]\text{.}$ Define $A:C\left(\left[a,b\right]\right)\to ℝ$ by $Af=f(t0).$ Show that $A$ is a bounded linear functional with $‖A‖=1.$
184. Let $T:{ℂ}^{m}\to {ℂ}^{n}$ be a linear transformation. Let $A$ be the matrix of $T$ and let $A*=A‾t.$
1. Show that the matrix of ${T}^{*}$ is ${A}^{*}\text{.}$
2. Show that $‖T‖=\sqrt{\gamma },$ where $\gamma$ is the largest eigenvalue of ${A}^{*}A\text{.}$
185. Let $V$ and $W$ be normed vector spaces and let $T:V\to W$ be a bounded linear operator. Show that ${T}^{*}T$ is self adjoint and positive.
186. Let $p\in {ℝ}_{\ge 1}$ and let $q$ be defined by $\frac{1}{p}+\frac{1}{q}=1\text{.}$ Show that the dual of the Banach space ${\ell }^{p}$ is ${\ell }^{q}\text{.}$
187. Let $p\in {ℝ}_{\ge 1}\text{.}$ Show that ${\ell }^{p}$ is a reflexive Banach space.
188. Let $X$ be a normed vector space and let $B=\left\{x\in X | ‖x‖\le 1\right\}\text{.}$ Let $T:X\to X$ be a compact operator. Show that $\stackrel{‾}{T\left(B\right)}$ is compact.
189. Let $X$ be a normed vector space and let $A$ be a bounded subset of $X\text{.}$ Let $T:X\to X$ be a compact operator. Show that $\stackrel{‾}{T\left(A\right)}$ is compact.
190. Let $X$ be a finite dimensional normed vector space and let $T:X\to X$ be a linear transformation. Show that $T$ is a compact operator.
191. Let $X=ℂ\left(\left[a,b\right]\right)$ be the space of continuous functions $f:\left[a,b\right]\to ℝ$ with the supremum norm. Let $k:\left[a,b\right]×\left[a,b\right]\to ℂ$ be a continuous function and define $T:X\to X$ 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 $T:H\to H$ be a bounded self adjoint operator. Show that $‖T‖=sup { |⟨Tx,x⟩| | x∈H,‖x‖=1 } .$
193. Let $H$ be a Hilbert space and let $T:H\to H$ be a nonzero compact self adjoint operator.
1. Show that there exists an eigenvalue $\lambda$ of $T$ such that $|\lambda |=‖T‖\text{.}$
2. Show that if $v$ is an eigenvector of $T$ with eigenvalue $\lambda$ such that $|\lambda |=‖T‖$ then $v$ is a solution of the extremal problem $max { ⟨Tu,u⟩ | u∈H,‖u‖ =1 } .$
194. Let $H$ be a Hilbert space and let $T:H\to H$ be a nonzero compact self adjoint operator. Show that there exists an orthonormal basis of eigenvectors for $H\text{.}$
195. Let $H$ be a Hilbert space and let $T:H\to H$ be a nonzero compact self adjoint operator. Let $\Lambda$ be the set of eigenvalues of $T\text{.}$ If $\mu \in \Lambda$ let $P\left(\mu \right)$ be the orthogonal projection onto the subspace ${X}_{\mu }$ of eigenvectors with eigenvalue $\mu \text{.}$ Show that if $x\in H$ then $Tx=\sum _{\mu \in \Lambda }\mu P\left(\mu \right)x\text{.}$
196. Let $V$ and $W$ be Banach spaces. Let $V\prime$ be the dual of $V$ and let $W\prime$ be the dual of $W\text{.}$ Let $T:V\to W$ be a bounded linear operator. Define ${T}^{*}:W\prime \to V\prime$ by $T*f=f∘T.$ 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 $V\text{.}$ Define $\phi :V\to V″$ by $(φ(x))(f)= f(x),forf∈ V′.$ Show that $\phi$ is injective.
198. Let $V$ and $W$ be reflexive Banach spaces and let $T:V\to W$ be a bounded linear operator.
1. Show that $T$ is transformed to ${{T}^{*}}^{*}$ by the isomorphisms $V\cong V″$ and $W\cong W″\text{.}$
2. Show that $‖{T}^{*}‖=‖T‖\text{.}$
3. Show that if $V$ and $W$ are Hilbert spaces then $T$ is transformed to ${T}^{*}$ by the natural isomorphisms $V\cong V\prime$ and $W\cong W\prime \text{.}$
199. Let $H$ be an infinite dimensional Hilbert space. Let $T:H\to H$ 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 $a,b\in ℝ$ with $a Let $\lambda \in ℝ$ and let $p:\left[a,b\right]\to {ℝ}_{>0}$ and $q:\left[a,b\right]\to ℝ$ with $p\in {C}^{1}\left(\left[a,b\right]\right)$ and $q\in {C}^{2}\left(\left[a,b\right]\right)\text{.}$ Let ${a}_{1},{a}_{2},{b}_{1},{b}_{2}\in ℝ$ with $\left({a}_{1},{a}_{2}\right)\ne \left(0,0\right)$ and $\left({b}_{1},{b}_{2}\right)\ne \left(0,0\right)\text{.}$ Let $L:{C}^{2}\left(\left[a,b\right]\right)\to C\left(\left[a,b\right]\right)$ be given by $Ly=(-py′)′+qy.$ Let $u,v\in {C}^{2}\left(\left[a,b\right]\right)$ such that $Lu=0,Lv=0, a1u(a)+a2u′ (a)=0,and b1v(b)+ b2v′(b)=0.$ Let $G:\left[a,b\right]×\left[a,b\right]\to ℝ$ be given by $G(s,t)= { v(t)u(s), if s≤t u(t)v(s), if t≤s.$ Define $T:{L}^{2}\left(\left[a,b\right]\right)\to {L}^{2}\left(\left[a,b\right]\right)$ by $(Tf)(t)= ∫abG(t,s)f(s) ds,for t∈[a,b].$
1. Show that the eigenvalues of $T$ are nonzero and each eigenvector $f$ satisfies ${a}_{1}f\left(a\right)+{a}_{2}f\prime \left(a\right)=0$ and ${b}_{1}f\left(b\right)+{b}_{2}f\prime \left(b\right)=0\text{.}$
2. Show that $f$ is an eigenvector of $T$ with eigenvalue $\mu$ if and only if $f$ is an eigenvector of $L$ with eigenvalue $\frac{1}{\mu }\text{.}$
3. Show that $L$ has a sequence of eigenvalues $\lambda \to \infty ,$ each eigenspace of $L$ is one dimensional and there is an orthonormal basis of ${L}^{2}\left(\left[a,b\right]\right)$ of eigenvectors of $L\text{.}$
201. Let $G:\left[0,\pi \right]×\left[0,\pi \right]\to ℝ$ be given by $G(t,s)= { t-stπ, if s≤t, s-stπ, if s≥t$ and let $T:{L}^{2}\left(\left[0,\pi \right]\right)\to {L}^{2}\left(\left[0,\pi \right]\right)$ be given by $(Tf)(t)=∫0π G(t,s)f(s)ds fort∈[0,π].$
1. Show that $T$ has eigenvalues ${\lambda }_{n}={n}^{2},$ $n\in {ℤ}_{>0},$ and corresponding eigenvectors ${s}_{n}\left(t\right)=\sqrt{\frac{2}{\pi }}\text{sin} nt\text{.}$
2. Show that the functions ${s}_{n}\left(t\right)=\sqrt{2\pi }\text{sin} nt,$ $n\in {ℤ}_{>0}$ form an orthonormal basis of ${L}^{2}\left(\left[0,\pi \right]\right)\text{.}$

Notes and References

This material is a conversion of the Metric and Hilbert Space notes of J. Hyam Rubinstein into a Homework question form. This is a way to create a curriculum for a course.