(AFr 4.4.1)
Let $X$ and $Y$ be normed linear spaces. A linear transformation
$T:\, X\to Y$ is continuous {\it iff} $T$ is continuous at one point.
Definition.
Let $X$ and $Y$ be normed linear spaces, $T:\, X\to Y$
and let there exists a constant $K>0$ s.t.
$$
\|Tx\| \le K\|x\| \qquad x\in X.
$$
Then $T$ is called a bounded linear map.
$$
\| T\|Ê= \sup_{x\in X} \frac{\|Tx\|}{\|x\|}.
$$
Theorem.
(AFr 4.4.2)
Let $X$, $Y$ be normed linear spaces. The operator $T:\, X\to Y$ is
continuous iff $T$ is bounded.
Definition.
By $\mathcal L(X,Y)$ we denote the space of all linear transformations
equipped with
a. $T+S$
b. $\lambda T$.
By $\mathcal B(X,Y)\subset \mathcal L(X,Y)$ we denote the set of all bounded
linear transformations.
Theorem.
(AFr 4.4.3)
Let $X$ and $Y$ be normed linear spaces. Then $\mathcal B(X,Y)$ is a
normed linear space with the norm
$$
\| T\|Ê= \sup_{x\in X} \frac{\|Tx\|}{\|x\|}.
$$
Definition.
The sequence $\{T_n\}_{n=1}^\infty$ of bounded operators
is said to be uniformly convergent if there exists bounded $T$ s.t.
$\|T_n - T\|\to 0$ as $n\to\infty$.
Theorem.
(AFr 4.4.4)
If $X$ is a normed linear space and $Y$ is a Banach space
then $\mathcal B(X,Y)$ is a Banach space.
Home exercises.
1. (ex. 4.4.4 from AFr)
Let $X$ be a Banach space and let
$f(z)=\sum_{n=0}^\infty a_n z^n$ be an entire function.
Prove that for every $T\in \mathcal B(X, X)$
$$
\sum_{n=0}^\infty a_n T^n
$$
is absolutely convergent in $\mathcal B(X):= \mathcal B(X, X)$.
2. (ex. 4.4.6 from AFr)
Find the norm of the operator $A\in \mathcal B(X)$given by
$$
Af(t) = t f(t), \qquad 0\le t\le 1,
$$
where {\bf a.} $X=C[0,1]$, {\bf b.} $X=L^p[0,1]$, $1\le p\le \infty$.
3. (ex. 4.4.9 from AFr)
Let
$$
Af(x) = \int_0^1 K(x,y) f(y) \,dy, \qquad x\in (0,1).
$$
Prove that if $K\in L^2((0,1)\times(0,1))$ then $A: \,L^2(0,1)\to L^2(0,1)$
is bounded.
4.
Let $\mathcal L$ be the Laplace transform defined by
$$
g(s) = \mathcal L f(s) = \int_0^\infty f(t) e^{-st}\,dt.
$$
Show that $\mathcal L:\, L^2(\Bbb R_+) \to L^2(\Bbb R_+)$ is bounded and
$$
\|\mathcal L\| \le \sqrt{\pi}.
$$
Tip: Write
$$
\int_0^\infty f(t) e^{-st}\,dt = \int_0^\infty f(t) t^{1/2}e^{-st/2}\,
t^{-1/2}e^{-st/2}\,dt
$$
and use Cauchy-Schwatz inequality.