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 $ℂ$ 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 $(ℂ,d)\text{.}$
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 $(X_1,d_1),\dots ,(X_n,d_n)$ be metric spaces. Define the product metric $d$ on $X_1\times \cdots \times X_n$ and show that $(X_1\times \cdots \times X_n,d)$ is a metric space.
5. Let $(X,\Vert \Vert )$ 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 ${\Vert \Vert }_p$ on ${ℝ}^n$ and show that $({ℝ}^n,{\Vert \Vert }_p)$ is a normed vector space.
9. Let $X$ be a nonempty set. Define the set of bounded functions $B(X,ℝ)$ and the sup norm on $B(X,ℝ)\text{.}$ Show that $B(X,ℝ),$ with this norm, is a normed vector space.
10. Let $a,b\in ℝ$ with $a<b\text{.}$ Define the set of continuous functions $C([a,b],ℝ)$ and the $L^1\text{-norm}$ on $C([a,b],ℝ)\text{.}$ Show that $C([a,b],ℝ),$ with this norm, is a normed vector space.
11. Let $a,b\in ℝ$ with $a<b\text{.}$ Show that the set $C_{\text{bd}}([a,b],ℝ)$ of bounded continuous functions is a metric subspace of $C([a,b],ℝ)$ 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 $(X_1,d_1),\dots ,(X_{\ell },d_{\ell })$ be metric spaces. Show that a sequence $\overline{x_n}=(x_n^{\left(1\right)},\dots ,x_n^{\left(\ell \right)})$ in $X_1\times \cdots \times X_{\ell }$ converges if and only if each of the sequences $x_n^{\left(i\right)}$ (in $X_i\text{)}$ converges.
14. Let $(X,d)$ be a metric space. Show that the metric $d\prime :X\times X\to ℝ$ given by $$d\prime (x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is equivalent to $d\text{.}$
15. Let $(X,d)$ be a metric space. Show that $(X,d\prime )$ is a bounded metric space, where $$d\prime (x,y)=\frac{d(x,y)}{1+d(x,y)}\text{.}$$
16. Give an example of $X$ and two metrics $d$ and $d\prime$ on $X$ such that $d$ is equivalent to $d\prime$ and $(X,d)$ is not bounded and $(X,d\prime )$ is bounded.
17. Let $(X_1,d_1),\dots ,(X_{\ell },d_{\ell })$ be metric spaces and let $(X_1\times \cdots \times X_{\ell },d)$ be the product metric space. Let $\sigma :(X_1\times \cdots \times X_{\ell })\times (X_1\times \cdots \times X_{\ell })\to ℝ$ given by $$\sigma (x,y)=\text{max}\left\{d_i(x_i,y_i)\hspace{0.17em}\right|\hspace{0.17em}1\le i\le \ell \}\text{.}$$ Show that $\sigma$ is a metric on $X_1\times \cdots \times X_{\ell }$ and $d$ is equivalent to $\sigma \text{.}$
18. Let $(X_1,d_1),\dots ,(X_{\ell },d_{\ell })$ be metric spaces and let $(X_1\times \cdots \times X_{\ell },d)$ be the product metric space. Let $\rho :(X_1\times \cdots \times X_{\ell })\times (X_1\times \cdots \times X_{\ell })\to ℝ$ be given by $$\rho (x,y)={\left(\sum _{i=1}^{\ell }{d}_i{(x_i,y_i)}^2\right)}^{\frac{1}{2}}\text{.}$$ Show that $\rho$ is a metric on $X_1\times \cdots \times X_{\ell }$ 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 $$d(x,y)\le kd\prime (x,y)\le k\prime d(x,y)\text{.}$$
20. Let $(X,d)$ 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 $\mathbb{Q}$ is open in the metric space topology on $(ℝ,d)$ and $\mathbb{Q}$ is not open in the metric space topology on $(ℝ,d\prime )\text{.}$
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 $\bar{E}$ be the closure of $E\text{.}$ Show that $E$ is closed if and only if $E=\bar{E}\text{.}$
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 $(X,d)$ be a metric space and let $x\in X$ and $r\in {ℝ}_{\text{<mo>></mo><mn>0</mn>}}\text{.}$ Show that the closed ball $$\bar{B}(x,r)=\{y\in X\hspace{0.17em}|\hspace{0.17em}d(x,y)\le r\}$$ is a closed set in the metric space topology on $X\text{.}$
29. Give an example of a metric space $(X,d)$ and a point $x\in X$ such that $\bar{B}(x,1)\ne \overline{B(x,1)}\text{.}$
30. Let $(X,d)$ be a metric space and let $x\in X$ and $r\in {ℝ}_{>0}\text{.}$ Show that $\overline{B(x,r)}\subseteq \bar{B}(x,r)\text{.}$
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. $[0,1)\subseteq ℝ$ is not open and not closed,
    2. $\mathbb{Q}\subseteq ℝ$ 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 $X_1\times \cdots \times X_{\ell }$ have the product topology. Show that
    1. If $A_1\subseteq X_1,\dots ,A_{\ell }\subseteq X_{\ell }$ are open then $A_1\times \cdots \times A_{\ell }\subseteq X_1\times \cdots \times X_{\ell }$ is open.
    2. If $F_1\subseteq X_1,\dots ,F_{\ell }\subseteq X_{\ell }$ are closed then $F_1\times \cdots \times F_{\ell }\subseteq X_1\times \cdots \times X_{\ell }$ is closed.
37. Let $(X_1,d_1),\dots ,(X_{\ell },d_{\ell })$ be metric spaces and let $d$ be the product metric on $X_1\times \cdots \times X_{\ell }\text{.}$ Show that the metric space topology on $(X_1\times \cdots \times X_{\ell },d)$ is the product topology for $X_1\times \cdots \times X_{\ell },$ 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 $B(x,r)\cap A\ne \varnothing$ and $B(x,r)\cap A^c\ne \varnothing$ 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) $\partial \left([0,1]\right)\text{.}$
    2. Determine $\partial \mathbb{Q}$ (with proof, of course).
41. Let $(X,d)$ 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 $(X,d)$ 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 $B(x,\epsilon )=\left\{x\right\}\text{.}$
43. Let $X=ℝ$ with the usual topology. Show that
    1. ${\mathbb{Z}}_{>0}\subseteq ℝ$ is a discrete set in $ℝ,$
    2. $\left\{\frac{1}{n}\hspace{0.17em}\right|\hspace{0.17em}n\in {\mathbb{Z}}_{>0}\}\subseteq ℝ$ 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 $\mathbb{Q}$ is dense in $ℝ\text{.}$
    2. Show that ${\mathbb{Q}}^c$ is dense in $ℝ\text{.}$
    3. Show that ${\mathbb{Z}}_{>0}$ is nowhere dense in $ℝ\text{.}$
    4. Show that $\mathbb{Z}$ 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 $(X,d)$ and $(Y,\rho )$ 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 $d(x,a)<\delta$ then $\rho (f\left(x\right),f\left(a\right))<\epsilon \text{.}$
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 $$f\left(B(a,\delta )\right)\subseteq B(f\left(a\right),\epsilon )\text{.}$$
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 $(X,d),(Y_1,{\rho }_1)$ and $(Y_2,{\rho }_2)$ be metric spaces. Let $f:X\to Y_1$ and $g:X\to Y_2$ be functions. Define $h:X\to Y_1\times Y_2$ by $h\left(x\right)=(f\left(x\right),g\left(x\right))\text{.}$ Let $a\in X\text{.}$ Show that $h$ is continuous at $a$ if and only if $f$ and $g$ are continuous at $a\text{.}$ $$\begin{array}{ccc}X& \stackrel{\Delta }{⟶}& X\times X\\ & & \downarrow f\times g\\ & & Y_1\times Y_2\end{array}\begin{array}{ccc}x& ⟼& (x,x)\\ & & ↧\\ & & (f\left(x\right),g\left(x\right))\end{array}$$
57. Let $(X,d),(Y_1,{\rho }_1)$ and $(Y_2,{\rho }_2)$ be metric spaces. Let $f:X\to Y_1$ and $g:X\to Y_2$ be functions. Define $h:X\to Y_1\times Y_2$ by $h\left(x\right)=(f\left(x\right),g\left(x\right))\text{.}$ Let $a\in X\text{.}$ Show that $h$ is continuous if and only if $f$ and $g$ are continuous.
58. Let $(X,d)$ be a metric space and let $f:X\to ℝ$ 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 $(X,d)$ be a metric space. Show that $d:X\times X\to ℝ$ is continuous.
61. Let $f:ℝ\times ℝ\to ℝ$ be given by $$f(x,y)=\{\begin{array}{cc}\frac{xy}{x^2+y^2},& \text{if}\hspace{0.17em}(x,y)\ne (0,0)\\ 0,& \text{if}\hspace{0.17em}(x,y)=(0,0)\text{.}\end{array}$$ If $a\in ℝ$ let ${\ell }_a:ℝ\to ℝ$ be given by ${\ell }_a\left(y\right)=f(a,y)\text{.}$ If $b\in ℝ$ let $r_b:ℝ\to ℝ$ be given by $r_b\left(x\right)=f(x,b)\text{.}$
    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 $(0,0)\text{.}$
62. Give an example of metric spaces $X,Y$ and $Z$ and a function $f:X\times Y\to \mathbb{Z}$ such that
    1. if $x\in X$ then $$\begin{array}{cccc}{\ell }_x:& Y& ⟶& Z\\ & y& ⟼& f(x,y)\end{array}$$ is continuous,
    2. if $y\in Y$ then $$\begin{array}{cccc}{r}_y:& X& ⟶& Z\\ & x& ⟼& f(x,y)\end{array}$$ is continuous,
    3. $f:X\times Y\to Z$ 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 $$h\left(x\right)=\{\begin{array}{cc}f\left(x\right),& \text{if}\hspace{0.17em}x\in A,\\ g\left(x\right),& \text{if}\hspace{0.17em}x\in B\text{.}\end{array}$$ Show that $h:X\to Y$ is continuous.
64. Show that the function $f:ℝ\to ℝ$ given by $$f\left(x\right)=\frac{x}{1+x^2},$$ 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 $(X,d)$ and $(Y,\rho )$ 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 $(X,d)$ and $(Y,\rho )$ 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 $\text{sup}\left\{\rho (f_k\left(x\right),f\left(x\right))\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\to 0\text{.}$
68. Let $\left\{f_k\right\}$ be a sequence of continuous functions from a metric space $(X,d)$ to a metric space $(Y,\rho )\text{.}$ Suppose that $\left\{f_k\right\}$ converges uniformly to $f:X\to Y\text{.}$ Show that $f:X\to Y$ is continuous.
69. Let $(X,d)$ be a metric space and let $x_1,x_2,\dots$ be a sequence in $X\text{.}$ Show that if $(x_1,x_2,\dots )$ is a Cauchy sequence then $\{x_1,x_2,\dots \}$ is bounded.
70. Let $(X,d)$ be a metric space and let $(x_1,x_2,\dots )$ be a sequence in $X\text{.}$ Show that if $(x_1,x_2,\dots )$ converges then $(x_1,x_2,\dots )$ is a Cauchy sequence.
71. Let $(X,d)$ be a metric space and let $(x_1,x_2,\dots )$ be a sequence in $X\text{.}$ Show that if $(x_1,x_2,\dots )$ is a Cauchy sequence and contains a convergent subsequence then $(x_1,x_2,\dots )$ converges.
72. Give an example of a metric space $(X,d)$ and a Cauchy sequence $(x_1,x_2,\dots )$ in $X$ that does not converge.
73. Give an example of a metric space $(X,d)$ that is not complete.
74. Show that $ℝ$ with the usual metric is a complete metric space.
75. Let $(X,d)$ be a complete metric space. Let $Y\subseteq X$ be a subspace of $X\text{.}$ Show that if $Y$ is closed then $(Y,d)$ is complete.
76. Give an example of a metric space $(X,d)$ and a subspace $Y\subseteq X$ 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 $Y\subseteq X$ be a subspace of $X\text{.}$ Show that if $(Y,d)$ is complete then $Y$ is a closed subset of $X\text{.}$
78. Let $(X_1,d_1),\dots ,(X_{\ell },d_{\ell })$ be metric spaces and let $(X_1\times \cdots \times X_{\ell },d)$ be the product metric space. Show that if $(X_1,d_1),\dots ,(X_{\ell },d_{\ell })$ are complete then $(X_1\times \cdots \times X_{\ell },d)$ is complete.
79. Let $(X,d)$ and $(Y,d\prime )$ be metric spaces and let $C_b(X,Y)$ be the set of bounded continuous functions $f:X\to Y$ with the metric given by $$\rho (f,g)=\text{sup}\left\{d\prime (f\left(x\right),g\left(x\right))\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\text{.}$$ Show that if $(Y,d\prime )$ is complete then $(C_b(X,y),\rho )$ is a complete metric space.
80. Let $(X,d)$ and $(Y,d\prime )$ be metric spaces and let $C_b(X,Y)$ be the set of bounded continuous functions $f:X\to Y$ with the metric $\rho :C_b(X,Y)\times C_b(X,Y)\to {ℝ}_{>0}$ given by $$\rho (f,g)=\text{sup}\left\{d\prime (f\left(x\right),g\left(x\right))\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\text{.}$$ Show that $(C_b(X,Y),\rho )$ is a metric space.
81. Let $(X,d)$ 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 $U=\mathbb{Q}$ and $V={\mathbb{Q}}^c\text{.}$ Show that $U$ and $V$ are dense and $U\cap B=\varnothing \text{.}$
83. Let $X=\mathbb{Q}$ with the usual metric and let $\mathbb{Q}=\{q_1,q_2,q_3,\dots \}$ be an enumeration of $\mathbb{Q}\text{.}$ For $n\in {\mathbb{Z}}_{>0}$ let $Q_n=\mathbb{Q}\backslash \left\{q_n\right\}\text{.}$
    1. Show that if $n\in {\mathbb{Z}}_{>0}$ then $Q_n$ is open and dense.
    2. Show that $\bigcap _{n\in {\mathbb{Z}}_{>0}}{Q}_n=\varnothing \text{.}$
84. Let $(X,d)$ be a complete metric space and let $\{U_1,U_2,U_3,\dots \}$ be a sequence of open and dense subsets of $X\text{.}$ show that $\bigcap _{n\in {\mathbb{Z}}_{>0}}{U}_n$ is dense in $X\text{.}$
85. Let $(X,d)$ be a complete metric space and let $\{F_1,F_2,F_3,\dots \}$ be a sequence of nowhere dense subsets of $X\text{.}$ Show that $\bigcup _{n\in {\mathbb{Z}}_{>0}}{F}_n$ 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 $X=\mathbb{Q},$ with the standard topology. Let $\mathbb{Q}=\{q_1,q_2,q_3,\dots \}$ be an enumeration of $\mathbb{Q}\text{.}$
    1. Show that $\left\{q_n\right\}$ is nowhere dense.
    2. Determine the interior of $\bigcup _{n\in {\mathbb{Z}}_{>0}}\left\{q_n\right\}\text{.}$
88. Let $(X,d)$ be a complete metric space and let $\{f_1,f_2,f_3,\dots \}$ be a sequence of continuous functions $$f_n:X\to ℝ,\text{for}\hspace{0.17em}n\in {\mathbb{Z}}_{>0}\text{.}$$ Assume that if $x\in X$ then $\{f_1\left(x\right),f_2\left(x\right),\dots \}$ 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 $n\in {\mathbb{Z}}_{>0}$ then $\left|f_n\left(x\right)\right|\le M\text{.}$
89. Show that the completion of $(0,1)$ with the usual metric is $[0,1]$ with the usual metric.
90. Let $(X,d)$ and $(Y,\rho )$ 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 $(X,d)$ 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 $((X_1,d_1),{\phi }_1)$ and $((X_2,d_2),{\phi }_2)$ be completions of $(X,d)\text{.}$ Show that there is a surjective isometry $f:X_1\to X_2$ such that $f\circ {\phi }_1={\phi }_2\text{.}$
95. Let $X=\{0,1\}$ and let $𝒯=\{\varnothing ,X,\left\{0\right\}\}\text{.}$
    1. Show $𝒯$ is a topology on $X\text{.}$
    2. Show that there does not exist a metric $d:X\times X\to {ℝ}_{\ge 0}$ such that $𝒯$ is the metric space topology of $(X,d)\text{.}$
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 $n\in {\mathbb{Z}}_{\ge 0}\text{.}$
    1. Show that the sequence $x_0,x_1,x_2,\dots$ converges in $X\text{.}$
    Let $p=\underset{n\to \infty }{\text{lim}}{x}_n\text{.}$
    2. Show that $d(x,p)\le \frac{d(x,f\left(x\right))}{1-\alpha }\text{.}$
    3. 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 $(x,y_1),(x,y_2)\in U$ then $|f(x,x_1)-f(x,y_2)|\le \alpha |y_1,y_2|\text{.}$ Show that if $(x_0,y_0)\in U$ then there exists $\delta \in {ℝ}_{>0}$ such that $y\prime \left(x\right)=f(x,y\left(x\right))$ has a unique solution $y:[x_0-\delta ,x_0+\delta ]\to ℝ$ such that $y\left(x_0\right)=y_0\text{.}$
99. Let $(X,\Vert \Vert )$ be a normed vector space. Show that $(X,\Vert \Vert )$ 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 $$\Vert f\Vert =\int \left|f\right|\text{and}d(f,g)=\Vert f-g\Vert$$ for $f,g\in S\text{.}$
     1. show that $\Vert \Vert :S\to {ℝ}_{\ge 0}$ is not a norm on $S\text{.}$
     2. Show that $d:S\times S\to {ℝ}_{\ge 0}$ 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 $\sum _{i\in {\mathbb{Z}}_{>0}}{f}_i$ be a series in $S$ which is norm absolutely convergent. Show that there exists a full set in ${ℝ}^k$ on which $\sum _{i\in {\mathbb{Z}}_{\ge 0}}{f}_i$ converges.
102. Let $S$ be the set of linear combinations of step functions $f:{ℝ}^k\to ℝ\text{.}$ Let $\sum _{n\in {\mathbb{Z}}_{>0}}{f}_k$ be a series in $S$ which is norm absolutely convergent. Show that $\sum _{n\in {\mathbb{Z}}_{>0}}{f}_n=0$ 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 $$\Vert f\Vert =\int f\text{and}d(f,g)=\Vert f-g\Vert$$ for $f,g\in L\prime \text{.}$
     1. Show that $\Vert \Vert :L\prime \to {ℝ}_{\ge 0}$ is a norm on $L\prime \text{.}$
     2. Show that $d:L\prime \times L\prime \to {ℝ}_{\ge 0}$ 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 $C([0,1],ℝ)=\{f:[0,1]\to ℝ\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is continuous}\}$ and let $$d(f,g)=\text{sup}\left\{|f\left(x\right)-g\left(x\right)|\hspace{0.17em}\right|\hspace{0.17em}x\in [0,1]\}\text{.}$$
     1. Show that $d:C([0,1],ℝ)\times C([0,1],ℝ)$ is a metric on $C([0,1],ℝ)\text{.}$
     2. Let $A={\bar{B}}_1\left(0\right)=\{f\in C([0,1],ℝ)\hspace{0.17em}|\hspace{0.17em}d(f,0)\le 1\}\text{.}$ 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 $(X,d)$ and $(Y,d\prime )$ 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 $$d(x,y)=\text{min}\{|x-y|,1\}\text{.}$$
     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 $\ell \in {\mathbb{Z}}_{>0}$ 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 $X=[0,2\pi )$ and $Y=S^1=\{(x,y)\in {ℝ}^2\hspace{0.17em}|\hspace{0.17em}{x}^2+y^2=1\}\text{.}$ Let $f:[0,2\pi )\to S^1$ be given by $$f\left(x\right)=(\text{cos}\hspace{0.17em}x,\text{sin}\hspace{0.17em}x)\text{.}$$
     1. Show that $f$ is continuous.
     2. Show that $f$ is a bijection.
     3. Show that $f^{-1}:S^1\to [0,2\pi )$ 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 $X_1-ℝ\backslash \left\{0\right\}$ and $X_2=\mathbb{Q}\text{.}$ Show that $X_1$ and $X_2$ 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 $f:X\to Y$ 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 $A,$ $\bar{A},$ is connected.
128. Let $A=(-\infty ,0)$ and $B=(0,\infty )$ 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 $X=\left\{(t,\text{sin}\left(\frac{\pi }{t}\right))\hspace{0.17em}\right|\hspace{0.17em}t\in (0,2]\}\text{.}$
     1. Let $\phi :{ℝ}^2\to ℝ$ be given by $\phi (x,y)=x\text{.}$ Show that $\phi :X\to (0,2]$ is a homeomorphism.
     2. Show that $X$ is connected.
     3. Show that $\bar{X}$ is connected.
     4. Show that $\bar{X}$ is not path connected.
139. Let $p\in {ℝ}_{\ge 1}\text{.}$ Let $V={ℝ}^n$ with $${\Vert (a_1,\dots ,a_n)\Vert }_p={({\left|a_1\right|}^p+\cdots +{\left|a_n\right|}^p)}^{\frac{1}{p}}\text{.}$$ Show that $(V,{\Vert \Vert }_p)$ is a Banach space.
140. Let $V={ℝ}^n$ with $${\Vert (a_1,\dots ,a_n)\Vert }_{\infty }=\text{sup}\{\left|a_1\right|,\left|a_2\right|,\dots ,\left|a_n\right|\}\text{.}$$ Show that $(V,{\Vert \Vert }_{\infty })$ is a Banach space.
141. Let ${\ell }^p$ be the vector space of sequences $(a_1,a_2,\dots )$ in $ℝ$ such that $\sum _{n\in {\mathbb{Z}}_{>0}}{\left|a_n\right|}^p<\infty \text{.}$ Let $${\Vert (a_1,a_2,\dots )\Vert }_q={\left(\sum _{n\in {\mathbb{Z}}_{>0}}{\left|a_n\right|}^p\right)}^{\frac{1}{p}}\text{.}$$ Show that $({\ell }^p,{\Vert \Vert }_p)$ is a Banach space.
142. Let ${\ell }^{\infty }$ be the vector space of bounded sequences $(a_1,a_2,\dots )$ in $ℝ$ with $${\Vert (a_1,a_2,\dots )\Vert }_{\infty }=\text{sup}\left\{\left|a_i\right|\hspace{0.17em}\right|\hspace{0.17em}i\in {\mathbb{Z}}_{>0}\}\text{.}$$ Show that $({\ell }^{\infty },{\Vert \Vert }_{\infty })$ is a Banach space.
143. Let $X$ be a topological space. Let $F=ℝ$ or $ℂ\text{.}$ Let $C_b(X,F)$ be the vector space of bounded continuous functions with $$\Vert f\Vert =\text{sup}\left\{\left|f\left(x\right)\right|\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\text{.}$$ Show that $(C_b(X,F),\Vert \Vert )$ is a Banach space.
144. Let $(V,\Vert \Vert )$ be a normed vector space. Show that $(V,\Vert \Vert )$ is a Banach space if and only if every norm absolutely convergent series is convergent.
145. Let $(V,\Vert \Vert )$ be a normed vector space. Show that a Schauder basis of $V$ is a total set.
146. Let $(V,\Vert \Vert )$ be a normed vector space. Show that if $V$ has a Schauder basis then $V$ is separable.
147. Let $(V,\Vert \Vert )$ 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 $e_i=(0,0,\dots ,0,1,0,0,\dots )$ with $1$ in the $i^{\text{th}}$ entry. Show that $\{e_1,e_2,e_3,\dots \}$ 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 $C_b([0,1],ℝ)$ be the vector space of bounded continuous functions on $[0,1]\text{.}$
     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 $C_b([0,1],ℝ)\text{.}$
     3. Show that $C_b([0,1],ℝ)$ is separable.
152. Let $(V,\Vert \Vert )$ 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 ${\Vert \Vert }_1$ and ${\Vert \Vert }_2$ be norms on $V\text{.}$ Show that ${\Vert \Vert }_1$ and ${\Vert \Vert }_2$ are equivalent.
154. Let $(V,\Vert \Vert )$ be a normed vector space. Show that if $V$ is finite dimensional then the closed unit ball in $V$ is compact.
155. Let $(V,\Vert \Vert )$ be an infinite dimensional Banach space. Construct a sequence $(e_1,e_2,\dots )$ of unit vectors in $V$ such that if $i,j\in {\mathbb{Z}}_{>0}$ and $i\ne j$ then $d(e_i,e_j)>\frac{1}{2}\text{.}$
156. Let $V={ℂ}^n$ with $$⟨(x_1,x_2,\dots ,x_n),(y_1,y_2,\dots ,y_n)⟩=\sum _{i=1}^n{x}_i\overline{y_i}\text{.}$$ show that $(V,⟨⟩)$ is a Hilbert space.
157. Let ${\ell }^2$ be the set of sequences $(a_1,a_2,\dots )$ in $ℂ$ such that $\sum _{i\in {\mathbb{Z}}_{>0}}{\left|a_i\right|}^2<\infty \text{.}$ Let $$⟨(x_1,x_2,\dots ),(y_1,y_2,\dots )⟩=\sum _{i\in {\mathbb{Z}}_{>0}}{x}_i\overline{y_i}\text{.}$$ Show that $({\ell }^2,⟨⟩)$ is a Hilbert space.
158. Carefully define the space $L^2\left([a,b]\right)$ and show that if $$⟨f,g⟩={\int }_a^{b}f\left(t\right)\overline{g\left(t\right)}dt$$ then $(L^2\left([a,b]\right),⟨,⟩)$ is a Hilbert space.
159. Let $(V,⟨⟩)$ be an inner product space and let $\Vert x\Vert =⟨x,x⟩$ for $x\in V\text{.}$ Show that if $x,y\in V$ then ${\Vert x+y\Vert }^2+{\Vert x-y\Vert }^2=2{\Vert x\Vert }^2+2{\Vert y\Vert }^2\text{.}$
160. Let $(V,\Vert \Vert )$ be a normed vector space. Define $$⟨x,y⟩=\frac{1}{4}({\Vert x+y\Vert }^2-{\Vert x-y\Vert }^2+i{\Vert x+iy\Vert }^2-i{\Vert x-iy\Vert }^2)$$ for $x,y\in V\text{.}$ Show that if $\Vert \Vert$ satisfies if $x,y\in V$ then ${\Vert x+y\Vert }^2+{\Vert x-y\Vert }^2=2{\Vert x\Vert }^2+2{\Vert y\Vert }^2$ then $⟨⟩$ is a norm on $V\text{.}$
161. Let $(V,⟨⟩)$ be an inner product space. Show that the Gram-Schmidt process produces an orthonormal basis of $V\text{.}$
162. Let $(V,⟨⟩)$ 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 $(V,⟨⟩)$ 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 $(V,⟨⟩)$ be a Hilbert space. Let $(a_1,a_2,\dots )$ be an orthonormal sequence in $V\text{.}$ Let $W=\text{span}\{a_1,a_2,\dots \}$ and let $M=\bar{W}$ be the closure of $W\text{.}$ Show that $P:V\to V$ given by $$P\left(x\right)=\sum _{n\in {\mathbb{Z}}_{>0}}⟨x,a_n⟩a_n$$ is an orthonormal projection onto $M\text{.}$
165. Let $(V,⟨⟩)$ be a Hilbert space. Let $(a_1,a_2,\dots )$ be an orthonormal sequence in $V\text{.}$ Let $x\in V\text{.}$ Show that $$P\left(x\right)=\sum _{n\in {\mathbb{Z}}_{>0}}⟨x,a_n⟩a_n$$ is independent of the order of the terms in the sum.
166. Let $(V,⟨⟩)$ be a Hilbert space. Let $(a_1,a_2,\dots )$ be an orthonormal sequence in $V\text{.}$ Let $x\in V\text{.}$ Show that $$\sum _{n\in {\mathbb{Z}}_{>0}}{\left|⟨x,a_n⟩\right|}^2\le {\Vert x\Vert }^2\text{.}$$
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 $\{a_1,a_2,\dots \}$ which is a total set. Show that $\{a_1,a_2,\dots \}$ is a Schauder basis for $V\text{.}$
169. Show that the functions $$e_m\left(t\right)=\frac{1}{\sqrt{2\pi }}{e}^{imt},\text{for}\hspace{0.17em}m\in \mathbb{Z},$$ form an orthonormal basis of $L^2\left([0,2\pi ]\right)\text{.}$
170. Let $(V,⟨⟩)$ 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 $V=C\left([0,1]\right)$ be the vector space of continuous functions $f:[0,1]\to ℝ$ with norm given by $$\Vert f\Vert ={\int }_0^1\left|f\left(t\right)\right|dt\text{.}$$ 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 $(a_1,a_2,\dots )$ in $ℝ$ with norm given by $$\Vert (a_1,a_2,\dots )\Vert =\text{sup}\{\left|a_1\right|,\left|a_2\right|,\dots \}\text{.}$$ Let $({\lambda }_1,{\lambda }_2,\dots )$ be a bounded sequence in $ℝ\text{.}$ Define $T:{\ell }^{\infty }\to {\ell }^{\infty }$ by $$T(a_1,a_2,\dots )=({\lambda }_1{a}_1,{\lambda }_2{a}_2,\dots )\text{.}$$
     1. Show that $T$ is a well defined linear transformation.
     2. Show that $$\Vert T\Vert =\text{sup}\{\left|{\lambda }_1\right|,\left|{\lambda }_2\right|,\dots \}\text{.}$$
177. Let $a,b\in ℝ$ with $a<b\text{.}$ Let $k:[a,b]\times [a,b]\to ℂ$ be a continuous function. Let $X=\{f:[a,b]\to ℂ\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is continuous}\}$ with the supremum norm. Define $T:X\to X$ by $\left(Tf\right)\left(t\right)={\int }_a^{b}k(t,s)x\left(s\right)ds\text{.}$
     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 $B(V,W)$ 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 $a,b\in ℝ$ with $a<b\text{.}$ Let $C\left([a,b]\right)$ be the vector space of continuous functions $f:[a,b]\to ℂ$ with the supremum norm. Define $T:C\left([a,b]\right)\to ℂ$ by $$Tf={\int }_a^{b}f\left(t\right)dt$$ Show that the operator norm of $T$ is $$\Vert T\Vert =b-a\text{.}$$
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 $${⟨Tx,y⟩}_2={⟨x,T^{*}y⟩}_1\text{.}$$
     2. Show that $T^{*}$ is a linear transformation.
     3. Show that $T^{*}$ is bounded.
     4. Show that $\Vert T^{*}\Vert =\Vert T\Vert \text{.}$
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 $${T^{*}}^{*}=T\text{.}$$
183. Let $a,b\in ℝ$ with $a<b\text{.}$ Let $C\left([a,b]\right)$ be the Banach space of continuous functions $f:[a,b]\to ℝ$ with the supremum norm. Let $t_0\in [a,b]\text{.}$ Define $A:C\left([a,b]\right)\to ℝ$ by $$Af=f\left(t_0\right)\text{.}$$ Show that $A$ is a bounded linear functional with $$\Vert A\Vert =1\text{.}$$
184. Let $T:{ℂ}^m\to {ℂ}^n$ be a linear transformation. Let $A$ be the matrix of $T$ and let $$A^{*}={\bar{A}}^t\text{.}$$
     1. Show that the matrix of $T^{*}$ is $A^{*}\text{.}$
     2. Show that $\Vert T\Vert =\sqrt{\gamma },$ 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 $B=\{x\in X\hspace{0.17em}|\hspace{0.17em}\Vert x\Vert \le 1\}\text{.}$ Let $T:X\to X$ be a compact operator. Show that $\overline{T\left(B\right)}$ 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 $\overline{T\left(A\right)}$ 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 $X=ℂ\left([a,b]\right)$ be the space of continuous functions $f:[a,b]\to ℝ$ with the supremum norm. Let $k:[a,b]\times [a,b]\to ℂ$ be a continuous function and define $T:X\to X$ by $$\left(Tf\right)\left(t\right)={\int }_a^{b}k(t,s)f\left(s\right)ds$$ 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 $$\Vert T\Vert =\text{sup}\left\{\left|⟨Tx,x⟩\right|\hspace{0.17em}\right|\hspace{0.17em}x\in H,\Vert x\Vert =1\}\text{.}$$
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 $\left|\lambda \right|=\Vert T\Vert \text{.}$
     2. Show that if $v$ is an eigenvector of $T$ with eigenvalue $\lambda$ such that $\left|\lambda \right|=\Vert T\Vert$ then $v$ is a solution of the extremal problem $$\text{max}\left\{⟨Tu,u⟩\hspace{0.17em}\right|\hspace{0.17em}u\in H,\Vert u\Vert =1\}\text{.}$$
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 $$T*f=f\circ T\text{.}$$ 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 $$\left(\phi \left(x\right)\right)\left(f\right)=f\left(x\right),\text{for}f\in V\prime \text{.}$$ 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 $\Vert T^{*}\Vert =\Vert T\Vert \text{.}$
     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 $a<b\text{.}$ Let $\lambda \in ℝ$ and let $p:[a,b]\to {ℝ}_{>0}$ and $q:[a,b]\to ℝ$ with $p\in C^1\left([a,b]\right)$ and $q\in C^2\left([a,b]\right)\text{.}$ Let $a_1,a_2,b_1,b_2\in ℝ$ with $(a_1,a_2)\ne (0,0)$ and $(b_1,b_2)\ne (0,0)\text{.}$ Let $L:C^2\left([a,b]\right)\to C\left([a,b]\right)$ be given by $$Ly=(-py\prime )\prime +qy\text{.}$$ Let $u,v\in C^2\left([a,b]\right)$ such that $$\begin{array}{c}Lu=0,Lv=0,\\ a_{1}u\left(a\right)+a_{2}u\prime \left(a\right)=0,\text{and}\\ b_{1}v\left(b\right)+b_{2}v\prime \left(b\right)=0\text{.}\end{array}$$ Let $G:[a,b]\times [a,b]\to ℝ$ be given by $$G(s,t)=\{\begin{array}{cc}v\left(t\right)u\left(s\right),& \text{if}\hspace{0.17em}s\le t\\ u\left(t\right)v\left(s\right),& \text{if}\hspace{0.17em}t\le s\text{.}\end{array}$$ Define $T:L^2\left([a,b]\right)\to L^2\left([a,b]\right)$ by $$\left(Tf\right)\left(t\right)={\int }_a^{b}G(t,s)f\left(s\right)ds,\text{for}\hspace{0.17em}t\in [a,b]\text{.}$$
     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 $L^2\left([a,b]\right)$ of eigenvectors of $L\text{.}$
201. Let $G:[0,\pi ]\times [0,\pi ]\to ℝ$ be given by $$G(t,s)=\{\begin{array}{cc}t-\frac{st}{\pi },& \text{if}\hspace{0.17em}s\le t,\\ s-\frac{st}{\pi },& \text{if}\hspace{0.17em}s\ge t\end{array}$$ and let $T:L^2\left([0,\pi ]\right)\to L^2\left([0,\pi ]\right)$ be given by $$\left(Tf\right)\left(t\right)={\int }_0^{\pi }G(t,s)f\left(s\right)ds\text{for}t\in [0,\pi ]\text{.}$$
     1. Show that $T$ has eigenvalues ${\lambda }_n=n^2,$ $n\in {\mathbb{Z}}_{>0},$ and corresponding eigenvectors $s_n\left(t\right)=\sqrt{\frac{2}{\pi }}\text{sin}\hspace{0.17em}nt\text{.}$
     2. Show that the functions $s_n\left(t\right)=\sqrt{2\pi }\text{sin}\hspace{0.17em}nt,$ $n\in {\mathbb{Z}}_{>0}$ form an orthonormal basis of $L^2\left([0,\pi ]\right)\text{.}$

Notes and References

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

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