(AFr 4.5.1) (Banach-Steinhaus theorem)
Let $X$ be a Banach space and $Y$ be a normed linear space.
Let $\{T_\alpha\}$ be a family of bounded linear operators
from $X$ to $Y$. It for each $x\in X$ the set $\{T_\alpha x\}$
is bounded, then the set $\{\|T_\alpha\|\}$ is bounded.
Definition.
Let $X$ and $Y$ be normed linear spaces and let $T_n:\, X\to Y$.
The sequence $\{T_n\}_{n=1}^\infty$ is said to be strongly convergent
if for any $x\in X$ the limit $\lim_{n\to\infty} T_n x$ exists for
any $x\in X$.
If there exists a bounded $T$ s.t.
$\lim_{n\to\infty} T_n x = Tx$ for any $x\in X$, then $\{T_n\}_{n=1}^\infty$
is called strongly convergent to $T$ ($T_n \to T$).
Theorem.
(AFr 4.5.2)
Let $X$ be a Banach space and $Y$ be a normed linear space and
let $\{T_n\}_{n=1}^\infty$ be the sequence of bounded operators.
If the sequence $\{T_n\}_{n=1}^\infty$ strongly convergent, then
there exists $T$ s.t. $T_n\to T$ strongly.
Theorem.
(AFr 4.6.1)(Open-mapping theorem)
Let $X$ and $Y$ be Banach spaces and let
$T:\, X\to Y$ be a mapping onto. Then $T$ maps open sets
of $X$ onto open sets of $Y$.
Home exercises.
1. (ex. 4.6.2 from AFr)
Let $X$ be a Banach space and let
$A\in \mathcal B(X)$, $\|A\|<1$. Show that $(I+A)^{-1}$ exists and
$$
(I+A)^{-1} = \sum_{n=0}^\infty (-1)^n A^n.
$$
2. (ex. 4.6.3 from AFr)
Let $X$ be a Banach space and let $T$ and $T^{-1}$ belong to
$\mathcal B(X)$. Show that is $S\mathcal B(X)$ and $\|S-T\| <1/\|T^{-1}\|$, then
$S^{-1}$ exists and
$$
\|S^{-1} - T^{-1}\| < \frac{\| T^{-1}\|}{1 - \|S-T\|\|T^{-1}\|}.
$$
3. (Holmgren)
Let
$$
Kf(x) = \int_0^1 K(x,y) f(y) \,dy, \qquad x\in (0,1).
$$
Assume that $K\in L^2(0,1)\to L^2 (0,1)$ is bounded.
Then
$$
\|K\| \le \Bigl(\sup_y\int_0^1|K(x,y)|\,dx\Bigr)^{1/2}
\Bigl(\sup_x\int_0^1|K(x,y)|\,dy\Bigr)^{1/2}.
$$