$\newcommand{\ip}[2]{\langle #1, #2 \rangle} \newcommand{\norm}[1]{\lVert #1 \rVert} \newcommand{\abs}[1]{\left| #1 \right|} \newcommand{\R}{\mathbb{R}} \newcommand{\ip}[2]{\langle #1, #2 \rangle} \DeclareMathOperator*{\range}{range} \DeclareMathOperator*{\kern}{kern} \DeclareMathOperator*{\det}{det} \DeclareMathOperator*{\diag}{diag} \DeclareMathOperator*{\trace}{Tr} \DeclareMathOperator*{\closure}{cl} $In this post we will explore some elementary facts on bounded linear operators in the context of normed vector spaces. The reference I am using for this is Kreyszig's Introductory Functional Analysis with Applications. Note the examples here come from Section 2.7.
Let's setup some definitions first. Let $(X, \norm{\cdot})$ denote a normed vector space. For simplicity, we assume all vector spaces are over the field $\R$. A linear operator $T : X \longrightarrow Y$ is a mapping from vector space $X$ to vector space $Y$ satisfying two properties
- For all $x, y \in X$, we have $T(x+y) = Tx + Ty$.
- For all $x \in X, a \in \R$, we have $T(ax) = aTx$.
To connect this with something more concrete, take the following example. If $X = \R^{n}$ and $Y = \R^{m}$, both equipped with the standard Euclidean $l_2$-norm, then from linear algebra we know that there exists an $A \in \R^{m,n}$ such that $Tx = Ax$ for all $x \in \R^{n}$. Using the definition of operator norm above, we see that $\norm{T} = \sigma_1(A)$, the top singular value of $A$.
Let's now explore some simple properties of bounded linear operators.
Fact. Here's an easy warm-up. Let $T_1, T_2$ be bounded linear operators,
where $T_2 : X \longrightarrow Y$ and $T_1 : Y \longrightarrow Z$. Define the composition operator
$T_1 \cdot T_2 : X \longrightarrow Z$ as $T_1( T_2 x) $. Then (a) $T_1 \cdot T_2$ is another
bounded linear operator and (b) $\norm{ T_1 \cdot T_2 } \leq \norm{T_1} \norm{T_2}$.
Proof. Linearity is trivial. To see boundedness, observe that for any $x \in X$,
\begin{align*}
\norm{ T_1 \cdot T_2 x } = \norm{T_1 (T_2 x)} \leq \norm{T_1}\norm{T_2 x} \leq \norm{T_1}\norm{T_2} x
\end{align*}
This yields both the rest of (a) and (b).
Proposition. Let $T$ be a bounded linear operator and suppose $T \neq 0$ (check that $T=0$ is a bounded linear operator).
Then for any $x \in X$ such that $\norm{x} < 1$, $\norm{Tx} < \norm{T}$.
Proof. Since $T \neq 0$, clearly $\norm{T} > 0$. If $x=0$ we are done, so suppose $x \neq 0$ with $\norm{x} < 1$.
Now if $\norm{T x} = 0$ we are also done, so assume otherwise.
Put $\alpha = 1/\norm{x}$. Then clearly $\alpha > 1$. Observe that,
\begin{align*}
\norm{Tx} < \alpha \norm{T x} = \norm{T (\alpha x)} \leq \norm{T} \norm{\alpha x} = \norm{T} \alpha \norm{x} = \norm{T}
\end{align*}
Proposition. Let $T : X \longrightarrow Y$ be a linear operator.
Then $T$ is continuous if and only if $T$ is bounded.
Proof. ($\Rightarrow$) Suppose $T$ is continuous. Then since $0 \in X$, $T$ is continuous at
$0$. By the definition of continuity, there exists a $\delta > 0$ such that for all $x \in X$ such
that $\norm{x} \leq \delta$, we have $\norm{ Tx } \leq 1$. In other words,
$$
\begin{align*}
\sup_{x \in X : \norm{x} \leq \delta} \norm{T x} &\leq 1
\end{align*}
$$
Put $\alpha = \frac{1}{\delta}$. Then
$$
\begin{align*}
\sup_{x \in X : \norm{x} \leq \delta} \norm{T x}
= \sup_{x \in X : \norm{\alpha x} \leq 1} \norm{T x}
= \sup_{x \in X : \norm{\alpha x} \leq 1} \frac{1}{\alpha} \norm{T (\alpha x)}
= \sup_{z \in X : \norm{z} \leq 1} \frac{1}{\alpha} \norm{T z}
\end{align*}
$$
Continuing the chain, we have
$$
\begin{align*}
\sup_{z \in X : \norm{z} \leq 1} \frac{1}{\alpha} \norm{T z}
\geq \sup_{z \in X : \norm{z} = 1} \frac{1}{\alpha} \norm{T z}
= \frac{1}{\alpha} \sup_{z \in X : \norm{z} = 1} \norm{T z}
\end{align*}
$$
And therefore $ \sup_{z \in X : \norm{z} = 1} \norm{T z} \leq \alpha < \infty$, which means $T$ is bounded.
($\Leftarrow$) Suppose $T$ is bounded. Fix $\epsilon > 0$ and $x_0 \in X$. Put $\delta = \epsilon/\norm{T}$. Then
observe for every $x \in X$ such that $\norm{ x - x_0 } \leq \delta$,
\begin{align*}
\norm{Tx - Tx_0} = \norm{T(x-x_0)} \leq \norm{T} \norm{x-x_0} \leq \norm{T} \delta \leq \epsilon
\end{align*}
Finally, we conclude with the following.
Proposition.
Let $T : D \longrightarrow Y$ be a bounded linear operator, where $D \subset X$ is a subspace of some
normed vector space, and $Y$ is a Banach space (complete normed vector space).
Then there exists another bounded linear operator $T_e : \closure(D) \longrightarrow Y$ such that
$T_e x = T x$ for all $x \in D$ and $\norm{T_e} = \norm{T}$, where $\closure(D)$ denotes the closure of $D$.
Proof. Assume $D \neq \closure(D)$, otherwise we are done.
Let $x \in \partial D$ (the boundary of $D$). This means there exists a sequence $(x_n)$ of points in $D$ such that
$x_n \rightarrow x$. For any $n,m$ we have
$$
\begin{align*}
\norm{T x_n - T x_m } = \norm{T (x_n - x_m)} \leq \norm{T} \norm{x_n - x_m}
\end{align*}
$$
and therefore, the sequence $(T x_n)$ is Cauchy since $(x_n)$ is Cauchy. But since $Y$ is complete, this means
$T x_n \rightarrow y$ for some $y \in Y$. Note that if another sequence $z_n \rightarrow y$, then
$T z_n \rightarrow y$ as well, by the continuity of $T$ (which we proved above). Therefore, given an $x \in \partial D$,
it makes sense to define $y(x) = \lim_{x_n \rightarrow x} Tx_n$.
Now define $T_e$ as follows:
$$
\begin{align*}
T_e x = \begin{cases}
T x & \text{if } x \in D \\
y(x) & \text{if } x \in \partial D
\end{cases}
\end{align*}
$$
Using the linearity of limits, it is straightforward to check that $T_e$ is indeed a linear operator on $\closure(D)$.
Boundedness comes from the definition of limit point (if $T_e$ were not bounded, then certainly $T$ would not be: by taking
any point arbitrarily close to the boundary we could make $\norm{Tx}$ arbitrarily large).
The final thing to show is that $\norm{T_e}=\norm{T}$. Certainly $\norm{T_e} \geq \norm{T}$ since
the supremum in the former is over $\closure(D) \supset D$ of the latter. Now let
$x \in \partial D$, and let $x_n \rightarrow x$. But since $\norm{T x_n} \leq \norm{T} \norm{x_n}$ for all $x_n$,
we have
$$
\begin{align*}
\norm{T x} = \lim_{x_n \rightarrow x} \norm{T x_n} \leq \norm{T} \lim_{x_n \rightarrow x} \norm{x_n} = \norm{T} \norm{x}
\end{align*}
$$
which holds since the functional $x \mapsto \norm{x}$ is continuous for any norm.
But this means $\norm{T_e} \geq \norm{T}$, which establishes the equality.
Note that implicitly in the proof of the last proposition we used the following fact about continuous functions, whose proof is immediate from the definitions.
Fact. Let $f : D \longrightarrow Y$ be a continuous function from some subset $D \subset X$, where $X$ and $Y$ are normed vector spaces. Then if $x_n \rightarrow x$ in $D$, then $f(x_n) \rightarrow f(x)$ in $Y$.