Let $T$ be a one-to-one linear operator $T:\, X\to Y$.
Then the inverse operator $T^{-1}$ is defined as
$$
T^{-1} y = x \qquad {\text if} \qquad Tx=y.
$$
Note that domain $T^{-1}$ is a linear subspace of $Y$.
Theorem.
(AFr 4.6.2)
Let $X$ and $Y$ be a Banach spaces and let $T$ be a one-to-one
bounded linear map from $X$ onto $Y$. Then $T^{-1}$ is a bounded map.
Corollary.
(AFr 4.6.3)
Let $X$ be Banach spaces equipped with either $\|\cdot\|_1$ or $\|\cdot\|_2$.
Suppose that there exists a constant $K$ s.t.
$$
\|x\|_1\le K\, \|x\|_2 \qquad {\text{ for all}} \qquad x\in X.
$$
Then there exists a constant $K'$ s.t.
$$
\|x\|_2\le K'\, \|x\|_1 \qquad {\text{ for all}} \qquad x\in X.
$$
Definition.
Let $X$ and $Y$ be linear vector spaces and let
$T:\, X\to Y$, where $T$ is defined on $D_T$. Then
$$
G_T = \{(x, Tx): \, x\in D_T\}
$$
is called the graph of $T$.
If $G_T$ is a closed set in $X\times Y$ then we say that $T$ is a closed operator.
Note that $T$ is closed iff
$$
x_n\in D_T \quad x_n\to x, \quad Tx_n \to y \Longrightarrow x\in D_T, \quad Tx=y.
$$
Theorem.
(AFr 4.6.4) (Closed graph theorem)
Let $X$ and $Y$ be Banach spaces and let $T:\, X\to Y$ be a linear operator
with $D_T = X$. If $T$ is closed then $T$ is continuous.
Definition.
Let $X$ and $Y$ be linear vector spaces and let
$T:\, X\to Y$ defined on $D_T$. A linear operator $S:\, X\to Y$
is called an extension of $T$ if $D_T\subset D_S$ and $Tx = Sx$ for all
$x\in D_T$.
Definition.
Let $X$ be a linear vector space and $Y$ be a normed space.
Let $T:\, X\to Y$ defined on $D_T$. Suppose that there exists $S$
such that
(i) $S$ is a closed linear operator
(ii) $S$ is an extension of $T$
(iii) If $S'$ satisfies (i) and (ii), then $S'$ is an extension of $S$.
Then we say that $S$ is a s closure of $T$. The closure of $T$
is denoted by $\overline T$.
Theorem.
(AFr 4.7.1)
Let $X$ and $Y$ be Banach spaces. $T$ has a closure $\overline T$
{\it iff}
$$
x_n\in D_T \quad x_n\to 0, \quad Tx_n \to y \Longrightarrow y=0.
$$