# Applications of Residues: Evaluaton of Improper Integrals 3 (Indented Paths)

Suppose that a function $f(z)$ has a simple pole at a point $z=x_0$ on the real axis, with a Laurent series representation in a punctured disk $0<|z-x_0|<R_2$ and with residue $B_0$. Let $C_\rho$ denote the upper half of a circle $|z-x_0|=\rho$, where $\rho<R_2$ and with the clockwise direction.

$f(z)$ can be written as
$$f(z)=g(z)+\frac{B_0}{z-x_0}\ (0<|z-x_0|<R_2),$$
where $g(z)=\sum_{n=0}^\infty a_n(z-x_0)^n$. So, we have
$$\int_{C_\rho}f(z)dz=\int_{C_\rho}g(z)dz+B_0\int_{C_\rho}\frac{dz}{z-x_0}.$$
Let $\rho<\rho_0<R_2$. Then $g(z)$ is bounded on $|z-x_0|\leq\rho_0$ i.e. there exists $M>0$ such that $|g(z)|\leq M$ whenever $|z-x_0|\leq\rho_0$. Thus, we get the estimate
$$\left|\int_{C_\rho}g(z)dz\right|\leq M\pi\rho.$$
Consequently, we have
$$\lim_{\rho\to 0}\int_{C_\rho} g(z)dz=0.$$
The semi-circle $-C_\rho$ has parametric representation
$$z=x_0+\rho e^{i\theta}\ (0\leq\theta\leq\pi).$$
Using this parametric representation, we calculate
\begin{align*}
\int_{C_\rho}\frac{dz}{z-x_0}&=-\int_{-C_{\rho}}\frac{dz}{z-x_0}\\
&=-\int_0^{\pi}\frac{1}{\rho e^{i\theta}}\rho i e^{i\theta}d\theta\\
&=-\pi i.
\end{align*}
Therefore, we obtain

\label{eq:indentpath}
\lim_{\rho\to 0}\int_{C_\rho}f(z)dz=-B_0\pi i.

Example. [Singularity on Contour of Integration] Evaluate the improper Integral
$$I=\int_0^\infty\frac{\sin x}{x}dx.$$

Solution. The function $f(z)=\frac{e^{iz}}{z}$ has a simple pole at $z=0$.

Since $f(z)$ is analytic within and on the simple closed contour, we have
$$\int_{C_R} \frac{e^{iz}}{z}dz+\int_{-R}^{-\rho}\frac{e^{ix}}{x}dx+\int_{C_\rho}\frac{e^{iz}}{z}dz+\int_{\rho}^R\frac{e^{ix}}{x}dx=0.$$
Note that $\frac{1}{z}$ is analytic at all points $z$ in the upper half plane that are exterior to the circle $C_\rho$ and that for any $z$ on $C_R$, $\left|\frac{1}{z}\right|=\frac{1}{R}$
and $\lim_{R\to\infty}\frac{1}{R}=0$. Thus, by Jordan’s Lemma
$$\lim_{R\to\infty}\int_{C_R}f(z)dz=\lim_{R\to\infty}\int_{C_R}\frac{e^{iz}}{z}dz=0.$$
As $R\to\infty$ and $\rho\to 0$, we obtain
$$\int_{-\infty}^\infty\frac{e^{ix}}{x}dx=-\lim_{\rho\to 0}\int_{C_\rho}\frac{e^{iz}}{z}dz=\pi i.$$
Therefore,
$$\int_0^\infty\frac{\sin x}{x}dx=\frac{\pi}{2}.$$

# Fresnel Integrals

In this lecture, we derive Fresnel integrals
$$\int_0^\infty\cos(x^2)dx=\int_0^\infty\sin(x^2)dx=\frac{1}{2}\sqrt{\frac{\pi}{2}},$$
which appear in optics and diffraction theory.

Let us consider a contour shown in the following figure.

As seen in the figure, $C_R$ is a part of the circle $z=Re^{i\theta}$, where $0\leq\theta\leq\frac{\pi}{4}$. Let $f(z)=e^{iz^2}$. Then $f(z)$ is analytic on and within the positively oriented simple closed contour $C$ shown in the figure. So, we have $\int_Cf(z)dz=0$ which amounts to the following expression:
$$\int_0^Re^{ix^2}dx+\int_{C_R}e^{iz^2}dz-\int_0^Re^{-r^2}\left(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\right)dr=0.$$
Separating this expression into the real and the imaginary parts, we obtain
\begin{align}
\label{eq:cos}
\int_0^R\cos(x^2)dx&=\frac{1}{\sqrt{2}}\int_0^Re^{-r^2}dr-\mathrm{Re}\int_{C_R}e^{iz^2}dz,\\
\label{eq:sin}
\int_0^R\sin(x^2)dx&=\frac{1}{\sqrt{2}}\int_0^Re^{-r^2}dr-\mathrm{Im}\int_{C_R}e^{iz^2}dz.
\end{align}
\begin{align*}
\left|\int_{C_R}e^{iz^2}dz\right|&=\left|\int_0^{\frac{\pi}{4}}e^{iR^2e^{2i\theta}}Rie^{i\theta}d\theta\right|\\
&\leq R\int_0^{\frac{\pi}{4}}|e^{iR^2e^{2i\theta}}|d\theta\\
&=R\int_0^{\frac{\pi}{4}}e^{-R^2\sin 2\theta}d\theta\\
&=\frac{R}{2}\int_0^{\frac{\pi}{2}}e^{-R^2\sin\phi}d\phi\ (\mbox{by subsitution}\ \phi=2\theta)\\
&=\frac{R}{4}\int_0^\pi e^{-R^2\sin\phi}d\phi\\
&<\frac{R}{4}\frac{\pi}{R^2}\\
&=\frac{\pi}{4R}\to 0
\end{align*}
as $R\to\infty$. The inequality in the second line to the last was obtained by Jordan’s Inequality. Hence, as $R\to\infty$ \eqref{eq:cos} and \eqref{eq:sin} become
\begin{align*}
\int_0^\infty\cos(x^2)dx&=\int_0^\infty\sin(x^2)dx\\
&=\frac{1}{\sqrt{2}}\int_0^\infty e^{-r^2}dr\\
&=\frac{1}{2}\sqrt{\frac{\pi}{2}}.
\end{align*}

# Jordan’s Lemma

Suppose that

1. a function $f(z)$ is analytic at all points $z$ in the upper half plane $y\geq 0$ that are exterior to a circle $|z|=R_0$.
2. For any $z$ on $C_R: |z|=R>R_0$, there exists a positive real number $M_R>0$ such that $|f(z)|\leq M_R$ and $\lim_{R\to\infty}M_R=0$.

Then for any positive real number $a$,
$$\lim_{R\to\infty}\int_{C_R}f(z)e^{iaz}dz=0.$$

Proof. We first show Jordan’s Inequality
$$\label{eq:jordan}\int_0^\pi e^{-R\sin\theta}d\theta<\frac{\pi}{R}\ (R>0).$$

The graphs of y=sin(theta) (in red) and y=(2/pi)theta (in blue)

As shown in the figure, $\frac{2}{\pi}\theta\leq\sin\theta$ for $0\leq\theta\leq\frac{\pi}{2}$. If $R>0$, then $$e^{-R\sin\theta}\leq e^{-2R\theta/\pi},\ 0\leq\theta\leq\frac{\pi}{2}.$$So, we have
\begin{align*}
\int_0^{\frac{\pi}{2}}e^{-R\sin\theta}d\theta&\leq\int_0^{\frac{\pi}{2}}e^{-2R\theta/\pi}d\theta\\
&=\frac{\pi}{2R}(1-e^{-R})\\
&<\frac{\pi}{2R}.
\end{align*}Since the graph of $y=\sin\theta$ is symmetric about $\theta=\frac{\pi}{2}$ on the interval $0\leq\theta\leq\pi$,
$$\int_0^\pi e^{-R\sin\theta}d\theta=2\int_0^{\frac{\pi}{2}}e^{-R\sin\theta}d\theta<\frac{\pi}{R}.$$

Let $C_R$ denote the positively oriented circle $z=Re^{i\theta}$ where $0\leq\theta\leq\pi$. Then
$$\int_{C_R}f(z)e^{iaz}dz=\int_0^\pi f(Re^{i\theta})\exp(iaRe^{i\theta})iRe^{i\theta}d\theta$$
and so
\begin{align*}
\left|\int_{C_R}f(z)e^{iaz}dz\right|&\leq M_RR\int_0^\pi e^{-aR\sin\theta}d\theta\\
&<\frac{M_R\pi}{a}\ (\mbox{by Jordan Inequality \eqref{eq:jordan}})\\
&\to 0
\end{align*}
as $R\to\infty$ since by assumption $\lim_{R\to\infty}M_R=0$.

# Harmonic Functions

Throughout this course, a connected open subset of $\mathbb{C}$ is called a domain. Suppose that a function $f(z)=u(x,y)+iv(x,y)$ is analytic in a domain $\mathcal{D}$. Then $f(z)$ satisfies the Cauchy-Riemann equations i.e.

\label{eq:c-r}
u_x=v_y,\ u_y=-v_x.

Differentiating the Cauchy-Riemann equations \eqref{eq:c-r} with respect to $x$, we obtain

\label{eq:c-r2}
u_{xx}=v_{yx},\ u_{yx}=-v_{xx}.

Differentiating the Cauchy-Riemann equations \eqref{eq:c-r} with respect to $y$, we obtain

\label{eq:c-r3}
u_{xy}=v_{yy},\ u_{yy}=-v_{xy}.

By the continuity of the partial derivatives of $u(x,y)$ and $v(x,y)$, we have

\label{eq:cont}
u_{xy}=u_{yx},\ v_{xy}=v_{yx}.

Applying \eqref{eq:cont} to \eqref{eq:c-r2} and \eqref{eq:c-r3}, we obtain the Laplace equations:
$$u_{xx}+u_{yy}=0,\ v_{xx}+v_{yy}=0.$$
That is to say, $u(x,y)$ and $v(x,y)$ are harmonic maps in $\mathcal{D}$.

Example. The function $f(z)=e^{-y}\sin x-ie^{-y}\cos x$ is entire (i.e analytic on the complex plane $\mathbb{C}$), so both $e^{-y}\sin x$ and $-e^{-y}\cos x$ are harmonic. (You can of course check it for yourself!)

If two functions $u(x,y)$, $v(x,y)$ are harmonic in a domain $\mathcal{D}$ and their first-order partial derivatives satisfy the Cauchy-Riemann equations \eqref{eq:c-r} throughout $\mathcal{D}$, $v(x,y)$ is said to be a harmonic conjugate of $u(x,y)$.

Theorem. A function $f(z)=u(x,y)+iv(x,y)$ is analytic in a domain $\mathcal{D}$ if and only if $v(x,y)$ is a harmonic conjugate of $u(x,y)$.

Remark. If $v(x,y)$ is a harmonic conjugate of $u(x,y)$ in some domain, it is not in general true that $u$ is a harmonic conjugate of $v$ there.

Example. Let $u(x,y)=x^2-y^2$ and $v(x,y)=2xy$. Since $f(z)=z^2=(x^2-y^2)+i2xy$ is entire, $v(x,y)$ is a harmonic conjugate of $u(x,y)$. However, $u(x,y)$ cannot be a harmonic conjugate of $v(x,y)$ since $2xy+i(x^2-y^2)$ is not analytic anywhere.

Example. [Finding a harmonic conjugate of a harmonic function] Let $u(x,y)=y^3-3x^2y$ and $v(x,y)$ be a harmonic conjugate of $u(x,y)$. Then it follows from the Cauchy-Riemann equations \eqref{eq:c-r} that $v_y=-6xy$. Integrating this with respect to $y$, we obtain
$$v(x,y)=-3xy^2+\phi(x),$$
where $\phi(x)$ is some unknown function. We determine $\phi(x)$ using $u_y=-v_x$:
$$v_x=-3y^2+\phi’(x).$$
Comparing this with $-u_y=-3y^2+3x^2$, we get $\phi’(x)=3x^2$ and so, $\phi(x)=x^3+C$ where $C$ is a constant. Hence, we find a harmonic conjugate of $u(x,y)$:
$$v(x,y)=-3xy^2+x^3+C,$$
where $C$ is a constant. The corresponding analytic function is
$$f(z)=(y^3-3x^2y)+i(-3xy^2+x^3+C),$$
where $C$ is a constant.

# Group Theory 13: Finitely Generated Abelian Groups

The group $\mathbb{Z}\times\mathbb{Z}_2$ is generated by $(1,0)$ and $(0,1)$. In general, the direct product of $n$ cyclic groups, each of which is either $\mathbb{Z}$ or $\mathbb{Z}_m$ is generated by $(1,0,\cdots,0)$, $(0,1,0,\cdots,0)$, $\cdots$, $(0,0,\cdots,0,1)$. Such a direct product may be generated by fewer elements. For example, $\mathbb{Z}_3\times\mathbb{Z}_4\times\mathbb{Z}_{35}$ is generated by the single element $(1,1,1)$ i.e. it is a cyclic group. Conversely, we have the following theorem holds.

Theorem. [Fundamental Theorem of Finitely Generated Abelian Groups] Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups in the form
$$\mathbb{Z}_{p_1^{r_1}}\times\mathbb{Z}_{p_2^{r_2}}\times\cdots\times\mathbb{Z}_{p_n^{r_n}}\times\mathbb{Z}\times\mathbb{Z}\times\cdots\times\mathbb{Z},$$where the $p_i$ are primes, not necessarily distinct and the $r_i$ are positive integers. The direct product is unique except for a possible rearrangement of the factors. The number of factors $\mathbb{Z}$ is called the Betti number of $G$.

Example. Find all abelian groups, up to isomorphism, of order 360.

Solution. $360=2^3\cdot 3^3\cdot 5$, so all abelian groups, up to isomorphism, of order 360 are
\begin{align*}
\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_3\times\mathbb{Z}_3\times\mathbb{Z}_5,\\
\mathbb{Z}_2\times\mathbb{Z}_4\times\mathbb{Z}_3\times\mathbb{Z}_3\times\mathbb{Z}_5,\\
\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_2\times\mathbb{Z}_9\times\mathbb{Z}_5,\\
\mathbb{Z}_2\times\mathbb{Z}_4\times\mathbb{Z}_9\times\mathbb{Z}_5,\\
\mathbb{Z}_8\times\mathbb{Z}_3\times\mathbb{Z}_3\times\times\mathbb{Z}_5,\\
\mathbb{Z}_8\times\mathbb{Z}_9\times\mathbb{Z}_5.
\end{align*}

Definition. A group $G$ is decomposable if it is isomorphic to a direct product of two proper nontrivial subgroups. Otherwise $G$ is indecomposable.

Theorem. The finite indecomposable abelian groups are exactly the cyclic groups with order a power of a prime.

Proof. If $G$ is a finite indecomposable abelian group, then by Fundamental Theorem of Finitely Generated Abelian Groups, $G$ is isomorphic to a direct product of cyclic groups of prime power order.Since $G$ is indecomposable, the direct product must consist of just one cyclic group of a prime power order. conversely, let $p$ be a prime. Then $\mathbb{Z}_{p^r}$ is indecomposable. If it were isomorphic to $\mathbb{Z}_{p^i}\times\mathbb{Z}_{p^j}$ where $i+j=r$, then every element has an order at most $p^{\max(i,j)}<p^r$.

Theorem. If $m$ divides the order of a finite abelian group $G$, then $G$ has a subgroup of order $m$.

Proof. Since $G$ is a finite abelian group,
$$G\cong\mathbb{Z}_{p_1^{r_1}}\times\mathbb{Z}_{p_2^{r_2}}\times\cdots\times\mathbb{Z}_{p_n^{r_n}}.$$Since $p_1^{r_1}p_2^{r_2}\cdots p_n^{r_n}=|G|$, $m$ must be of the form $p_1^{s_1}p_2^{s_2}\cdots p_n^{s_n}$, where $0\leq s_i\leq r_i$. For each $1\leq i\leq n$, $p_i^{r_i-s_i}$ generates a cyclic group of order$$\frac{p_i^{r_i}}{(p_i^{r_i},p_i^{r_i-s_i})}=\frac{p_i^{r_i}}{p_i^{r_i-s_i}}=p_i^{s_i}.$$So, $p_i^{r_i-s_i}$ generates a cyclic subgroup of $\mathbb{Z}_{p_i^{r_i}}$ of order $p_i^{s_i}$. Therefore,$$\langle p_1^{r_1-s_1}\rangle\times\langle p_2^{r_2-s_2}\rangle\times\cdots\times\langle p_n^{r_n-s_n}\rangle$$
is a subgroup of $G$ of order $m=p_1^{s_1}p_2^{s_2}\cdots p_n^{s_n}$.

Theorem. If $m$ is a square free integer i.e. if $m$ is not divisible by the square of any prime, then every abelian group of order $m$ is cyclic.

Proof. Let $G$ be an abelian group of square free order $m$. Then
$$G\cong\mathbb{Z}_{p_1^{r_1}}\times\mathbb{Z}_{p_2^{r_2}}\times\cdots\times\mathbb{Z}_{p_n^{r_n}},$$
where $m=p_1^{r_1}p_2^{r_2}\cdots p_n^{r_n}$. Since $m$ is square free, $r_i=1$, $i=1,\cdots,n$ and the $p_i$ are distinct. Hence, $G$ is isomorphic to $\mathbb{Z}_{p_1p_2\cdots p_n}$ i.e. it is cyclic.

# Group Theory 12: Direct Products

Let $G_1,G_2,\cdots,G_n$ be groups. Consider the Cartesian product
$$\prod_{i=1}^nG_i:=G_1\times G_2\times\cdots\times G_n.$$
Define a binary operation $\cdot$ on $\prod_{i=1}^nG_i$ by
$$(a_1,a_2,\cdots,a_n)\cdot(b_1,b_2,\cdots,b_n)=(a_1b_1,a_2b_2,\cdots,a_nb_n)$$
for $(a_1,a_2,\cdots,a_n),(b_1,b_2,\cdots,b_n)\in\prod_{i=1}^nG_i$. Then $\cdot$ is well-defined. Clearly $\cdot$ is associative. $(e_1,e_2,\cdots,e_n)\in\prod_{i=1}^nG_i$ is an identity element. For each $(a_1,a_2,\cdots,a_n)\in\prod_{i=1}^nG_i$, $(a_1,a_2,\cdots,a_n)^{-1}=(a_1^{-1},a_2^{-1},\cdots,a_n^{-1})\in\prod_{i=1}^nG_i$. So, $\left(\prod_{i=1}^nG_i,\cdot\right)$ is a group called the direct product of the $G_i$’s. If the operation on each $G_i$ is commutative, refer to $\prod_{i=1}^nG_i$ as the direct sum of the groups $G_i$. In this case, we often use the notation $\bigoplus_{i=1}^nG_i$ instead of $\prod_{i=1}^nG_i$.

Example. Let $\mathbb{R}\oplus\mathbb{R}$ be the direct sum of $(\mathbb{R},+)$ and itself. Define a map $\varphi:\mathbb{R}\oplus\mathbb{R}\longrightarrow S^1\times S^1$ by
$$\varphi(x,y)=(e^{2\pi ix},e^{2\pi iy})$$
for each $(x,y)\in\mathbb{R}\oplus\mathbb{R}$. Then $\varphi$ is an onto-homomorphism. The kernel of $\varphi$ is
$$\ker\varphi=\mathbb{Z}\oplus\mathbb{Z}.$$
Hence, by the Fundamental Homomorphism Theorem
$$\mathbb{R}\oplus\mathbb{R}/\mathbb{Z}\oplus\mathbb{Z}\cong S^1\times S^1.$$That is, the quotient group $\mathbb{R}\oplus\mathbb{R}/\mathbb{Z}\oplus\mathbb{Z}$ is a torus. $\mathbb{R}\oplus\mathbb{R}/\mathbb{Z}\oplus\mathbb{Z}$ can be viwed as a quotient set $\mathbb{R}\oplus\mathbb{R}/\sim$ where $\sim$ is an equivalence relation on $\mathbb{R}\oplus\mathbb{R}$ defined as follows: For all $(x,y),(z,w)\in\mathbb{R}\oplus\mathbb{R}$,$$(x,y)\sim (z,w)\ \mbox{if}\ (x,y)-(z,w)=(m,n)$$for some $(m,n)\in\mathbb{Z}\times\mathbb{Z}$.

The following theorem is introduced without a proof.

Theorem. Let $(a_1,\cdots,a_n)\in\prod_{i=1}^nG_i$. If for each $i=1,\cdots,n$, $a_i$ is of finite order $r_i$ in $G_i$, then the order of $(a_1,\cdots,a_n)$ in $\prod_{i=1}^nG_i$ is the least common multiple of $r_1,r_2,\cdots,r_n$.

Example. Find the order of $(8,4,10)$ in $\mathbb{Z}_{12}\times\mathbb{Z}_{60}\times\mathbb{Z}_{24}$.

Solution. First we find the orders of 8, 4, 10 in $\mathbb{Z}_{12}$, $\mathbb{Z}_{60}$, and $\mathbb{Z}_{24}$, respectively. For that let us recall a theorem we studied here. The theorem can be restated for an additive group as:

Theorem. Let $G$ be a finite additive group and $a\in G$ with $|a|=n$. Then for any $k\in\mathbb{Z}$,

1.  $|ka|=\frac{|a|}{(k,|a|)}$.
2.  $|ka|=n$ if and only if $(k,|a|)=1$.

Since 1 has order $n$ in $\mathbb{Z}_n$, we have the following corollary.

Corollary. The order of $1\leq k\leq n-1$ in $\mathbb{Z}_n$ is $\frac{n}{(k,n)}$.

It follows from this corollary that the order of 8 in $\mathbb{Z}_{12}$ is $\frac{12}{(8,12)}=\frac{12}{4}=3$, the order of 4 in $\mathbb{Z}_{60}$ is $\frac{60}{(4,60)}=\frac{60}{4}=15$, and the order of 10 in $\mathbb{Z}_{24}$ is $\frac{24}{(10,24)}=\frac{24}{2}=12$. The least common multiple of 3, 15, 12 is 60, so the order of $(8,4,10)$ in $\mathbb{Z}_{12}\times\mathbb{Z}_{60}\times\mathbb{Z}_{24}$ is 60.

Example. $\mathbb{Z}_2\times\mathbb{Z}_3=\{(0,0), (0,1), (0,2),(1,0),(1,1),(1,3)\}$ is a cyclic group generated by $(1,1)$. Hence, $\mathbb{Z}_2\times\mathbb{Z}_3\cong\mathbb{Z}_6$.

Example. $\mathbb{Z}_2\times\mathbb{Z}_2$ is not cyclic. $\mathbb{Z}_2\times\mathbb{Z}_2\cong V_4$, Klein four-group.

Theorem. $\mathbb{Z}_m\times\mathbb{Z}_n$ is cyclic and isomorphic to $\mathbb{Z}_{mn}$ if and only if $(m,n)=1$.

Corollary. $\prod_{i=1}^n\mathbb{Z}_{m_i}$ is cyclic and isomorphic to $\mathbb{Z}_{m_1m_2\cdots m_n}$ if and only if the numbers $m_i$ for $i=1,2,\cdots,n$ are such that the greatest common divisor of any two of them is 1.

Corollary. If $n=(p_1)^{n_1}(p_2)^{n_2}\cdots(p_r)^{n_r}$ where $p_1,p_2,\cdots,p_r$ are distinct primes, then
$$\mathbb{Z}_n\cong\mathbb{Z}_{(p_1)^{n_1}}\times\mathbb{Z}_{(p_2)^{n_2}}\times\cdots\times\mathbb{Z}_{(p_r)^{n_r}}.$$

Example. $\mathbb{Z}_8\times\mathbb{Z}_9\cong\mathbb{Z}_{72}$.

Let $\prod_{i=1}^nG_i$ be the direct product of groups $G_1,\cdots,G_n$. For each $i=1,\cdots,n$, let
$$\bar G_i=\{(e_1,e_2,\cdots,e_{i-1},a_i,e_{i+1},\dots,e_n): a_i\in G_i\}\leq\prod_{i=1}^n G_i.$$
Then for each $i=1,\cdots,n$, $\bar G_i\cong G_i$. The direct product $\prod_{i=1}^n\bar G_i$ of the groups $\bar G_1,\cdots,\bar G_n$ is called an internal direct product while $\prod_{i=1}^nG_i$ is called an external direct product. Clearly the external and internal direct products are isomorphic to each other.

# Group Theory 11: The Isomorphism Theorems

The following theorem is called the Fundamental Homomorphism Theorem or the First Isomorphism Theorem.

Theorem. Let $G$ and $G’$ be groups, and $\varphi:G\longrightarrow G’$ an epimorphism (onto homomorphism). Then $G/K\cong G’$ where $K=\ker\varphi$.

Proof.

Let $\gamma: G\longrightarrow G/K$ be the canonical homomorphism i.e. for any $a\in G$, $\gamma(a)=Ka$. Define $\psi: G/K\longrightarrow G’$ by
$$\psi(Ka)=\varphi(a)$$for each $a\in G$. Then

1. $\psi$ is well-defined:\begin{align*}Ka=Kb&\Rightarrow ab^{-1}\in K\\&\Rightarrow\varphi(ab^{-1})=e’\\&\Rightarrow\psi(Ka)=\varphi(a)=\varphi(b)=\psi(Kb).\end{align*}
2. $\psi$ is a homomorphism:$$\psi(KaKb)=\psi(Kab)=\varphi(ab)=\varphi(a)\varphi(b)=\psi(Ka)\psi(Kb).$$
3. $\psi$ is one-to-one:\begin{align*}\psi(Ka)=\psi(Kb)&\Rightarrow\varphi(a)=\varphi(b)\\&\Rightarrow\varphi(ab^{-1})=e’\\&\Rightarrow ab^{-1}\in K\\&\Rightarrow Ka=Kb.\end{align*}
4. $\psi$ is onto: Let $b\in G’$. Then there exists $a\in G$ such that $\varphi(a)=b$ since $\varphi$ is onto. Now, $Ka\in G/K$ and $\psi(Ka)=\varphi(a)=b$.

Example. Let $S^1$ be the unit circle centered at the origin. Then it can be represented in terms of complex numbers as
$$S^1=\{e^{2x\pi i}:x\in[0,1)\}.$$Define a map $\varphi: (\mathbb{R},+)\longrightarrow(S^1,\cdot)$ by $$\varphi(x)=e^{2\pi ix}$$for each $x\in\mathbb{R}$. Then $\varphi$ is an onto homomorphism. The kernel of $\varphi$ is$$\ker\varphi=\mathbb{Z}.$$Hence, by the Fundamental Homomorphism Theorem
$$\mathbb{R}/\mathbb{Z}\cong S^1.$$Note that $\mathbb{R}/\mathbb{Z}$ can be viewed as the quotient set $\mathbb{R}/\sim$ where $\sim$ is an equivalence relation on $\mathbb{R}$ defined as follows: For all $x,y\in\mathbb{R}$,
$$x\sim y\ \mbox{if}\ x-y=n$$for some $n\in\mathbb{Z}$.

Theorem [Correspondence Theorem]. Let $\varphi: G\longrightarrow G’$ be a homomorphism. Let $K=\ker\varphi$, $H’\leq G’$ and $H=\varphi^{-1}(H’)=\{a\in G: \varphi(a)\in H\}$. Then $K\subset H\leq G$ and $H/K\cong H’$. If $H’\triangleleft G’$, then $H\triangleleft G’$.

Proof. Since $e\in K\subset H$, $H\ne\emptyset$. Let $a,b\in H$. Then $\varphi(a),\varphi(b)\in H’$. Since $H’\leq G’$, $\varphi(a)\varphi(b)^{-1}=\varphi(ab^{-1})\in H’$ and so, $ab^{-1}\in H$. Hence, $H\leq G$. Since $e’\in H’$, $K\subset H$. Let $\psi=\varphi|_H$. Then $\psi$ is a homomorphism from $H$ onto $H’$ and $\ker\psi=\ker\varphi=K$. Therefore, by the Fundamental Homomorphism Theorem $H/K\cong H’$.

Suppose that $H’\triangleleft G’$. Let $a\in G$ and $h\in H$. Then $\varphi(a)\in G’$ and $\varphi(h)\in H’$. Since $H’\triangleleft G’$, $\varphi(a)\varphi(h)\varphi(a)^{-1}=\varphi(aha^{-1})\in H’$ which implies that $aha^{-1}\in H$. Hence, $H\triangleleft G$.

Theorem [The Second Isomorphism Theorem]. Let $H\leq G$ and $N\triangleleft G$. Then $HN\leq G$, $H\cap N\triangleleft G$ and
$$H/H\cap N\cong HN/N.$$

Proof. Let $a,b\in HN$. Then $a=h_1n_1$ and $b=h_2n_2$ for some $h_1,h_2\in H$ and $n_1,n_2\in N$. Then
\begin{align*}
ab^{-1}&=h_1n_1(h_2n_2)^{-1}\\
&=h_1n_1(n_2^{-1}h_2^{-1})\\
&=h_1h_2^{-1}h_2(n_1n_2^{-1})h_2^{-1})\in HN.
\end{align*}
So, $HN\leq G$. Clearly $H\cap N\triangleleft G$, in particular $H\cap N\triangleleft H$. Also clearly $N\leq HN$. Let $n_1\in N$. Then $\forall hn\in HN$,
$$(hn)n_1(hn)^{-1}=h(nn_1n^{-1})h^{-1}\in N.$$This means that $N\triangleleft HN$. Define $\varphi:H\longrightarrow HN/N$ by
$$\varphi(h)=Nh$$for each $h\in H$. Then $\varphi$ is clearly well-defined, a homomorphism. Let $Nhn\in HN/N$. Then $Nhn=hnN=hN=Nh$ and $\varphi(h)=Nh=Nhn$. So, $\varphi$ is onto.
\begin{align*}
h\in \ker\varphi&\Leftrightarrow \varphi(h)=Nh=N\\
&\Leftrightarrow h\in N\\
&\Leftrightarrow h\in H\cap N.
\end{align*}
So, $\ker\varphi=H\cap N$. Therefore, by the Fundamental Homomorphism Theorem
$$H/H\cap N\cong HN/N.$$

Theorem [The Third Isomorphism Theorem]. Let $\varphi: G\longrightarrow G’$ be an epimorphism. Let $K=\ker\varphi$, $N’\triangleleft G’$, and $N=\varphi^{-1}(N’)=\{a\in G: \varphi(a)\in N’\}$. Then
$$G/N\cong G’/N,$$or equivalently
$$G/N\cong(G/K)/(N/K).$$

Proof. Define $\psi: G\longrightarrow G’/N’$ by
$$\psi(a)=N’\varphi(a)$$for each $a\in G$. Then

1. $\psi$ is well-defined:\begin{align*}a=b\in G &\Rightarrow\varphi(a)=\varphi(b)\\&\Rightarrow \psi(a)=N’\varphi(a)=N’\varphi(b)=\psi(b).\end{align*}
2. $\psi$ is a homomorphism:\begin{align*}\psi(ab)&=N’\varphi(ab)\\&=N’\varphi(a)\varphi(b)\\&=N’\varphi(a)N’\varphi(b)\\&=\psi(a)\psi(b).\end{align*}
3. $\psi$ is onto: Let $N’c\in G’/N’$. Then $c\in G’$. Since $\varphi$ is onto, there exists $a\in G$ such that $\varphi(a)=c$. $N’c=N’\varphi(a)=\psi(a)$.
4. $\ker\psi=N$:\begin{align*}a\in\ker\psi&\Leftrightarrow N’\varphi(a)=N’\\&\Leftrightarrow\varphi(a)\in N’\\&\Leftrightarrow a\in N.\end{align*}

Therefore, by the Fundamental Homomorphism Theorem we obtain
$$G/N\cong G’/N’.$$Since $\varphi:G\longrightarrow G’$ is an epimorphism, by the Fundamental Homomorphism Theorem, $G’\cong G/K$. Since $N=\varphi^{-1}(N’)$, by the Correspondence Theorem, $N’\cong N/K$. Hence, $G’/N’\cong(G/K)/(N/K)$ i.e. $G/N\cong(G/K)/(N/K)$.

# Functional Analysis 10: Linear Functionals

Definition. Let $X$ be a vector space. A linear functional is a linear map $f:\mathcal{D}(f)\subset X\longrightarrow\mathbb{R}$ (or $f:\mathcal{D}(f)\subset X\longrightarrow\mathbb{C}$).

Definition. A linear functional $f:\mathcal{D}(f)\longrightarrow\mathbb{R}$ is said to be bounded if there exists a number $c$ such that $|f(x)|\leq c||x||$ for all $x\in\mathcal{D}(f)$. Just as in linear operators case $||f||$ is defined by
\begin{align*}
||f||&=\sup_{\begin{array}{c}x\in\mathcal{D}(f)\\x\ne O\end{array}}\frac{|f(x)|}{||x||}\\
&=\sup_{\begin{array}{c}x\in\mathcal{D}(f)\\||x||=1\end{array}}|f(x)|.
\end{align*}
Also we have the inequality holds
$$|f(x)|\leq ||f||||x||$$
for all $x\in\mathcal{D}(f)$.

Just as in linear operators case, we have the following theorem holds.

Theorem. A linear functional $f$ with domain $\mathcal{D}(f)$ in a normed space is continuous if and only if $f$ is bounded.

Example. Let $a=(\alpha_j)\in\mathbb{R}^3$. Define $f:\mathbb{R}^3\longrightarrow\mathbb{R}$ by
$$f(x)=x\cdot a=\xi_1\alpha_1+\xi_2\alpha_2+\xi_3\alpha_3$$ for each $x=(\xi_j)\in\mathbb{R}^3$. Then $f$ is a linear functional. By Cauchy-Schwarz inequality, we obtain
$$|f(x)|=|x\cdot a|\leq ||x||||a||$$
which implies $||f||\leq ||a||$. On the other hand, for $x=a$
$$||a||=\frac{||a||^2}{||a||}=\frac{|f(a)|}{||a||}\leq ||f||.$$
Hence, we have $||f||=||a||$.

Example. Define $f:\mathcal{C}[a,b]\longrightarrow\mathbb{R}$ by
$$f(x)=\int_a^b x(t)dt$$
for each $x(t)\in\mathcal{C}[a,b]$. Then $f$ is a linear functional.
\begin{align*}
|f(x)|&\leq\left|\int_a^b x(t)dt\right|\\
&\leq(b-a)\max|x(t)|\\
&=(b-a)||x||.
\end{align*}
So, $||f||\leq b-a$. Let $x=x_0=1$. Then
\begin{align*}
b-a&=\int_a^b dt\\
&=\frac{|f(x_0)|}{||x_0||}\\
&\leq ||f||.
\end{align*}
Hence, we have $||f||=b-a$.

Let $X^\ast$ be the set of all linear functionals. Then $X\ast$ can be made into a vector space. For any $f,g\in X^\ast$ and scalar $\alpha$, define addition $f+g$ and scalar multiplication $\alpha f$ as follows: For each $x\in X$,
\begin{align*}
(f+g)(x)&=f(x)+g(x),\\
(\alpha f)(x)&=\alpha f(x).
\end{align*}
$X^\ast$ is called the dual space of $X$. One may also consider $X^{\ast\ast}=(X^\ast)^\ast$, the dual space of $X^\ast$. Fix $x\in X$. Define a map $g_x: X^\ast\longrightarrow\mathbb{R}$ by
$$g_x(f)=f(x)$$
for each $f\in X^\ast$. For any $f_1,f_2\in X^\ast$,
\begin{align*}
f_1=f_2&\Longrightarrow f_1(x)=f_2(x)\\
&\Longrightarrow g_x(f_1)=g_x(f_2).
\end{align*}
so, $g_x$ is well-defined. Furthermore, $g_x$ is linear. To show this, for any $f_1,f_2\in X^\ast$ and scalars $\alpha,\beta$,
\begin{align*}
g_x(\alpha f_1+\beta f_2)&=(\alpha f_1+\beta f_2)(x)\\
&=\alpha f_1(x)+\beta f_2(x)\\
&=\alpha g_x(f_1)+\beta g_x(f_2).
\end{align*}
Define a map $C: X\longrightarrow X^{\ast\ast}$ by
$$Cx=g_x$$
for each $x\in X$. Then $C$ is a linear map. First let $x=y\in X$. Then for any $f\in X^\ast$, $g_x(f)=f(x)=f(y)=g_y(f)$, so $C(x)=g_x=g_y=C(y)$. Hence, $C$ is well-defined. To show that $C$ is linear, let $x,y\in X$ and $\alpha,\beta$ scalars. For any $f\in X^\ast$,
\begin{align*}
g_{\alpha x+\beta y}(f)&=f(\alpha x+\beta y)\\
&=\alpha f(x)+\beta f(y)\ (f\ \mbox{is linear})\\
&=\alpha g_x(f)+\beta g_y(f)\\
&=(\alpha g_x+\beta g_y)(f).
\end{align*}
Thus,
$$C(\alpha x+\beta y)=g_{\alpha x+\beta y}=\alpha g_x+\beta g_y=\alpha Cx+\beta Cy.$$
If $X$ is an inner product space or $X$ is a finite dimensional vector space, $C$ becomes oen-to-one. Let us assume that $X$ is equipped with an inner product $\langle\ ,\ \rangle$. Then for any fixed $a\in X$, the map $f_a: X\longrightarrow\mathbb{R}$ defined by
$$f_a(x)=\langle a,x\rangle\ \mbox{for each}\ x\in X$$
is a linear functional. Let $Cx=Cy$. Then $g_{x-y}=0$ and so $g_{x-y}(f_{x-y})=||x-y||^2=0$, hence $x=y$. Therefore, $C$ is one-to-one. We will discussed the case when $X$ is finite dimensional in the next lecture. If $C$ is one-to-one, $X$ is embedded into $X^{\ast\ast}$. We call $C:X\hookrightarrow X^{\ast\ast}$ the canonical embedding. (Here, the notation $\hookrightarrow$ means an embedding or a monomorphism.) If in addition $C$ is onto i.e. $X\stackrel{C}{\cong}X^{\ast\ast}$, then $X$ is said to be algebraically reflexive. If $X$ is finite dimensional, then $X$ is algebraically reflexive. This will be discussed in the next lecture as well.

# Functional Analysis 9: Bounded and Continuous Linear Operators

Definition. Let $X,Y$ be normed spaces and $T:\mathcal{D}(T)\longrightarrow Y$ be a linear operator where $\mathcal{D}(T)\subset X$. $T$ is said to be bounded if there exists $c\in\mathbb{R}$ such that for any $x\in\mathcal{D}(T)$,
$$||Tx||\leq c||x||.$$

Suppose that $x\ne O$. Then
$$\frac{||Tx||}{||x||}\leq c.$$
Let
$$||T||:=\sup_{\begin{array}{c}x\in\mathcal{D}(T)\\ x\ne O\end{array}}\frac{||Tx||}{||x||}.$$
Then $||T||$ is called the norm of the operator $T$. If $\mathcal{D}(T)=\{O\}$ then we define $||T||=0$.

Lemma. Let $T$ be a bounded linear operator. Then

1. $||T||=\displaystyle\sup_{\begin{array}{c}x\in\mathcal{D}(T)\\||x||=1\end{array}}||Tx||.$
2. $||\cdot||$ defined on bounded linear operators satisfies (N1)-(N3).

Proof.

1. \begin{align*}||T||&=\sup_{\begin{array}{c}x\in\mathcal{D}(T)\\x\ne O\end{array}}\frac{||Tx||}{||x||}\frac{||Tx||}{||x||}\\&=\sup_{\begin{array}{c}x\in\mathcal{D}(T)\\x\ne O\end{array}}\left|\left|\frac{||Tx||}{||x||}\right|\right|\\&=\sup_{\begin{array}{c}y\in\mathcal{D}(T)\\||y||=1\end{array}}||Ty||.\end{align*}
2. \begin{align*}||T||=0&\Longleftrightarrow Tx=0,\ \forall x\in\mathcal{D}(T)\\&\Longleftrightarrow T=0.\end{align*} Since $$\sup_{\begin{array}{c}x\in\mathcal{D}(T)\\||x||=1\end{array}}||(T_1+T_2)x||\leq \sup_{\begin{array}{c}x\in\mathcal{D}(T)\\||x||=1\end{array}}||T_1x||+\sup_{\begin{array}{c}x\in\mathcal{D}(T)\\||x||=1\end{array}}||T_2x||,$$ $$||T_1+T_2||\leq ||T_1||+||T_2||.$$

Examples.

1. The identity operator $I:X\longrightarrow X$ with $X\ne\{O\}$ is a bounded linear operator with $||I||=1$.
2. Zero operator $O: X\longrightarrow Y$ is a bounded linear operator with $||O||=0$.
3. Let $X$ be the normed space of all polynomials on $[0,1]$ with $||x||=\max_{t\in[0,1]}|x(t)|$. Differentiation$$T: X\longrightarrow X;\ Tx(t)=x’(t)$$ is not a bounded operator. To see this, let $x_n(t)=t^n$, $n\in\mathbb{N}$. Then $||x_n||=1$ for all $n\in\mathbb{N}$. $Tx_n(t)=nt^{n-1}$ and $||Tx_n||=n$, for all $n\in\mathbb{N}$. So, $\frac{||Tx_n||}{||x_n||}=n$ and hence $||T||$ is not bounded.
4. Integral operator $$T:\mathcal{C}[0,1]\longrightarrow\mathcal{C}[0,1];\ Tx=\int_0^1\kappa(t,\tau)x(\tau)d\tau$$ is a bounded linear operator. The function $\kappa(t,\tau)$ is a continuous function on $[0,1]\times[0,1]$ called the kernel of $T$. \begin{align*}||Tx||&=\max_{t\in[0,1]}\left|\int_0^1\kappa(t,\tau)x(\tau)d\tau\right|\\&\leq\max_{t\in[0,1]}\int_0^1|\kappa(t,\tau)||x(\tau)|d\tau\\&\leq k_0||x||,\end{align*}where $k_0=\displaystyle\max_{(t,\tau)\in[0,1]\times[0,1]}\kappa(t,\tau)$.
5. Let $A=(\alpha_{jk})$ be an $r\times n$ matrix of real entries. The linear map $T:\mathbb{R}^n\longrightarrow\mathbb{R}^r$ given by $Tx=Ax$ for each $x\in\mathbb{R}^n$ is bounded. To see this, Let $x\in\mathbb{R}^n$ and write $x=(\xi_j)$. Then $||x||=\sqrt{\displaystyle\sum_{m=1}^n\xi_m^2}$.\begin{align*}||Tx||^2&=\sum_{j=1}^r\left[\sum_{k=1}^n\alpha_{jk}\xi_k\right]^2\\&\leq\sum_{j=1}^r\left[\left(\sum_{k=1}^n\alpha_{jk}^2\right)^\frac{1}{2}\left(\sum_{m=1}^n\xi_m\right)^\frac{1}{2}\right]^2\\&=||x||^2\sum_{j=1}^r\sum_{k=1}^n\alpha_{jk}^2.\end{align*}By setting $c^2=\displaystyle\sum_{j=1}^r\sum_{k=1}^n\alpha_{jk}^2$, we obtain$$||Tx||^2\leq c^2||x||^2.$$

In general, if a normed space $X$ is finite dimensional, then every linear operator on $X$ is bounded. Before we discuss this, we first introduce the following lemma without proof.

Lemma. Let $\{x_1,\cdots,x_n\}$ be a linearly independent set of vectors in a normed space $X$. Then there exist a number $c>0$ such that for any scalars $\alpha_1,\cdots,\alpha_n$, we have the inequality
$$||\alpha_1x_1+\cdots+\alpha_nx_n||\geq c(|\alpha_1|+\cdots+|\alpha_n|).$$

Theorem. If a normed space $X$ is finite dimensional, then every linear operator on $X$ is bounded.

Proof. Let $\dim X=n$ and $\{e_1,\cdots,e_n\}$ be a basis for $X$. Let $x=\displaystyle\sum_{j=1}^n\xi_je_j\in X$. Then
\begin{align*}
||Tx||&=||\sum_{j=1}^n\xi_jTe_j||\\
&\leq\sum_{j=1}^n||\xi_j|||Te_j||\\
&\leq\max_{k=1,\cdots,n}||Te_k||\sum_{j=1}^n|\xi_j|.
\end{align*}
By Lemma, there exists a number $c>0$ such that
$$||x||=||\xi_1e_1+\cdots+\xi_ne_n||\geq c(|\xi_1|+\cdots+|\xi_n|)=c\sum_{j=1}^n|\xi_j|.$$
So, $\displaystyle\sum_{j=1}^n|\xi_j|\leq\frac{1}{c}||x||$ and hence
$$||Tx||\leq M||x||,$$ where
$M=\frac{1}{c}\max_{k=1,\cdots,n}||Te_k||$.

What is really nice about linear operators from a normed space into a normed space is that a linear operator being bounded is equivalent to it being continuous.

Theorem. Let $X,Y$ be normed spaces and $T:\mathcal{D}(T)\subset X\longrightarrow Y$ a linear operator. Then

1. $T$ is continuous if and only if $T$ is bounded.
2. If $T$ is continuous at a single point, it is continuous.

Proof.

1. If $T=O$, then we are done. Suppose that $T\ne O$. Then $||T||\ne 0$. Assume that $T$ is bounded and $x_0\in\mathcal{D}(T)$. Let $\epsilon>0$ be given. Choose $\delta=\frac{\epsilon}{||T||}$. Then for any $x\in\mathcal{D}(T)$ such that $||x-x_0||<\delta$, $$||Tx-Tx_0||=||T(x-x_0)||\leq ||T||||x-x_0||<\epsilon.$$ Conversely, assume that $T$ is continuous at $x_0\in\mathcal{D}(T)$. Then given $\epsilon>0$ there exists $\delta>0$ such that $||Tx-Tx_0||<\epsilon$ whenever $||x-x_0||\leq\delta$. Take $y\ne 0\in\mathcal{D}(T)$ and set $$x=x_0+\frac{\delta}{||y||}y.$$ Then $x-x_0=\frac{\delta}{||y||}y$ and $||x-x_0||=\delta$. So,\begin{align*}||Tx-Tx_0||&=||T(x-x_0)||\\&=\left|\left|T\left(\frac{\delta}{||y||}y\right)\right|\right|\\&=\frac{\delta}{||y||}||Ty||\\&<\epsilon.\end{align*}Hence, for any $y\in\mathcal{D}(T)$, $||Ty||\leq\frac{\epsilon}{\delta}||y||$ i.e. $T$ is bounded.
2. In the proof of part (a), we have shown that if $T$ is continuous at a point, it is bounded. If $T$ is bounded, then it is continuous by part (a).

Corollary. Let $T$ be a bounded linear operator. Then

1. If $x_n\to x$ then $Tx_n\to Tx$.
2. $\mathcal{N}(T)$ is closed.

Proof.

1. If $T$ is bounded, it is continuous and so the statement is true.
2. Let $x\in\overline{\mathcal{N}(T)}$. Then there exists a sequence $(x_n)\subset\mathcal{N}(T)$ such that $x_n\to x$. Since $Tx_n=0$ for each $n=1,2,\cdots$, $Tx=0$. Hence, $x\in\mathcal{N}(T)$.

Theorem. Let $X$ be a normed space and $Y$ a Banach space. Let $T:\mathcal{D}(T)\subset X\longrightarrow Y$ be a bounded linear operator. Then $T$ has an extension $\tilde T:\overline{\mathcal{D}(T)}\longrightarrow Y$ where $\tilde T$ is a bounded linear operator of norm $||\tilde T||=||T||$.

Proof. Let $x\in\overline{\mathcal{D}(T)}$. Then there exists a sequence $(x_n)\subset\mathcal{D}(T)$ such that $x_n\to x$. Since $T$ is bounded and linear,
\begin{align*}
||Tx_m-Tx_n||&=||T(x_m-x_n)||\\
&\leq||T||||x_m-x_n||,
\end{align*}
for all $m,n\in\mathbb{N}$. Since $(x_n)$ is convergent, it is Cauchy so given $\epsilon>0$ there exists a positive integer $N$ such that for all $m,n\geq N$, $||x_m-x_n||<\frac{\epsilon}{||T||}$. Hence, for all $m,n>N$,
\begin{align*}
||Tx_m-Tx_n||&\leq ||T||||x_m-x_n||\\
&<\epsilon.
\end{align*}
That is, $(Tx_n)$ is a Cauchy sequence in $Y$. Since $Y$ is a Banach space, there exists $y\in Y$ such that $Tx_n\to y$. Define $\tilde T:\overline{\mathcal{D}(T)}\longrightarrow Y$ by $\tilde Tx=y$. In order for $\tilde T$ to be well- defined, its definition should not depend on the choice $(x_n)$. Suppose that there is a sequence $(z_n)\subset\mathcal{D}(T)$ such that $z_n\to x$. Then $x_n-z_n\to 0$. Since $T$ is bounded, it is continuous so $T(x_n-z_n)\to 0$. This means that $\displaystyle\lim_{n\to\infty}Tz_n=\lim_{n\to\infty}Tx_n=y$. $\tilde T$ is linear and $\tilde T|_{\mathcal{D}(T)}=T$. To show that $\tilde T$ is bounded, let $x\in\overline{\mathcal{D}(T)}$. Then there exists a sequence $(x_n)\subset\mathcal{D}(T)$ such that $x_n\to x$ as before. Since $T$ is bounded, for each $n=1,2,\cdots$,
$$||Tx_n||\leq ||T||||x_n||.$$ Since the norm $x\longmapsto||x||$ is continuous, as $n\to\infty$ we obtain
$$||\tilde Tx||\leq ||T||||x||.$$ Hence, $\tilde T$ is bounded and $||\tilde T||\leq ||T||$. On the other hand, since $\tilde T$ is an extension of $T$, $||T||\leq||\tilde T||$. Therefore, $||\tilde T||=||T||$.

# Functional Analysis 8: Linear Operators

From here on, a map from a vector space into another vector space will be called an operator.

Definition. A linear operator $T$ is an operator such that

1. $T(x+y)=Tx+Ty$ for any two vectors $x$ and $y$.
2. $T(\alpha x)=\alpha Tx$ for any vector $x$ and a scalar $\alpha$.

Proposition. An operator $T$ is a linear operator if and only if
$$T(\alpha x+\beta y)=\alpha Tx+\beta Ty$$
for any vectors $x,y$ and scalars $\alpha,\beta$.

Denote by $\mathcal{D}(T)$, $\mathcal{R}(T)$ and $\mathcal{N}(T)$, the domain, the range and the null space, respectively, of a linear operator $T$. The null space $\mathcal{N}(T)$ is the kernel of $T$ i.e.
$$\mathcal{N}(T)=T^{-1}(0)=\{x\in \mathcal{D}(T): Tx=0\}.$$
Since the term kernel is reserved for something else in functional analysis, we call it the null space of $T$.

Example. [Differentiation] Let $X$ be the space of all polynomials on $[a,b]$. Define an operator $T: X\longrightarrow X$ by
$$Tx(t)=x’(t)$$
for each $x(t)\in X$. Then $T$ is linear and onto.

Example. [Integration] Recall that $\mathcal{C}[a,b]$ denotes the space of all continuous functions on the closed interval $[a,b]$. Define an operator $T:\mathcal{C}[a,b]\longrightarrow\mathcal{C}[a,b]$ by
$$Tx(t)=\int_a^tx(\tau)d\tau$$
for each $x(t)\in\mathcal{C}[a,b]$. Then $T$ is linear.

Example. Let $A=(a_{jk})$ be an $r\times n$ matrix of real entries. Define an operator $T: \mathbb{R}^n\longrightarrow\mathbb{R}^r$ by
$$Tx=Ax=(a_{jk})(\xi_l)=\left(\sum_{k=1}^na_{jk}\xi_k\right)$$
for each $n\times 1$ column vector $x=(\xi_l)\in\mathbb{R}^n$. Then $T$ is linear as seen in linear algebra.

Theorem. Let $T$ be a linear operators. Then

1. The range $\mathcal{R}(T)$ is a vector space.
2. If $\dim\mathcal{D}(T)=n<\infty$, then $\dim\mathcal{R}\leq n$.
3. The null space $\mathcal{N}(T)$ is a vector space.

Proof. Parts 1 and 3 are straightforward. We prove part 2. Choose $y_1,\cdots,y_{n+1}\in\mathcal{R}(T)$. Then $y_1=Tx_1,\cdots,y_{n+1}=Tx_{n+1}$ for some $x_1,\cdots,\\x_{n+1}\in\mathcal{D}(T)$. Since $\dim\mathcal{D}(T)=n$, $x_1,\cdots,x_{n+1}$ are linearly dependent. So, there exist scalars $\alpha_1,\cdots,\alpha_{n+1}$ not all equal to 0 such that $\alpha_1x_1+\cdots+\alpha_{n+1}x_{n+1}=0$. Since $T(\alpha_1x_1+\cdots+\alpha_{n+1}x_{n+1})=\alpha_1y_1+\cdots+\alpha_{n+1}y_{n+1}=0$, $\mathcal{R}$ has no linearly independent subset of $n+1$ or more elements.

Theorem. $T$ is one-to-one if and only if $\mathcal{N}=\{O\}$.

Proof. Suppose that $T$ is one-to-one. Let $a\in\mathcal{N}$. Then $Ta=O=TO$. Since $T$ is one-to-one, $a=O$ and hence $\mathcal{N}=\{O\}$. Suppose that $\mathcal{N}=\{O\}$. Let $Ta=Tb$. Then by linearity $T(a-b)=O$ and so $a-b\in\mathcal{N}=\{O\}\Longrightarrow a=b$. Thus, $T$ is one-to-one.
Theorem.

1. $T^{-1}: \mathcal{R}(T)\longrightarrow\mathcal{D}(T)$ exists if and only if $\mathcal{N}=\{O\}$ if and only if $T$ is one-to-one.
2. If $T^{-1}$ exists, it is linear.
3. If $\dim\mathcal{D}(T)=n<\infty$ and $T^{-1}$ exists, then $\dim\mathcal{R}(T)=\dim\mathcal{D}(T)$.

Proof. Part 1 is trivial. Part 3 follows from part 2 of the previous theorem. Let us prove part 2. Let $y_1,y_2\in\mathcal{R}(T)$. Then there exist $x_1,x_2\in\mathcal{D}(T)$ such that $y_1=Tx_1$, $y_2=Tx_2$. Now,
\begin{align*}
\alpha y_1+\beta y_2&=\alpha Tx_1+\beta Tx_2\\
&=T(\alpha x_1+\beta x_2).
\end{align*}
So,
\begin{align*}
T^{-1}(\alpha y_1+\beta y_2)&=T^{-1}(T(\alpha x_1+\beta x_2))\\
&=\alpha x_1+\beta x_2\\
&=\alpha T^{-1}y_1+\beta T^{-1}y_2.
\end{align*}