> *Definition 1*: a **linear operator** $T$ is a linear mapping such that
>
> 1. the domain $\mathscr{D}(T)$ of $T$ is a vector space and the range $\mathscr{R}(T)$ of $T$ is contained in a vector space over the same field as $\mathscr{D}(T)$.
Injectivity of $T$ is equivalent to $\mathscr{N}(T) = \{0\}$.
??? note "*Proof*:"
Will be added later.
> *Theorem 1*: if a linear operator $T: \mathscr{D}(T) \to \mathscr{R}(T)$ is injective there exists a mapping $T^{-1}: \mathscr{R}(T) \to \mathscr{D}(T)$ such that
>
> $$
> y = Tx \iff T^{-1} y = x,
> $$
>
> for all $x \in \mathscr{D}(T)$, denoted as the **inverse operator**.
??? note "*Proof*:"
Will be added later.
> *Proposition 3*: let $T: \mathscr{D}(T) \to \mathscr{R}(T)$ be an injective linear operator, if $\mathscr{D}(T)$ is finite-dimensional, then
>
> $$
> \dim \mathscr{D}(T) = \dim \mathscr{R}(T).
> $$
??? note "*Proof*:"
Will be added later.
> *Lemma 1*: let $X,Y$ and $Z$ be vector spaces and let $T: X \to Y$ and $S: Y \to Z$ be injective linear operators, then $(ST)^{-1}: Z \to X$ exists and
>
> $$
> (ST)^{-1} = T^{-1} S^{-1}.
> $$
??? note "*Proof*:"
Will be added later.
We finish this subsection with a definition of the space of linear operators.
> *Definition 3*: let $\mathscr{L}(X,Y)$ denote the set of linear operators mapping from a vector space $X$ to a vector space $Y$.
From this definition the following theorem follows.
> *Theorem 2*: let $X$ and $Y$ be vectors spaces, the set of linear operators $\mathscr{L}(X,Y)$ is a vector space.
??? note "*Proof*:"
Will be added later.
Therefore, we may also call $\mathscr{L}(X,Y)$ the space of linear operators.
## Bounded linear operators
> *Definition 4*: let $(X, \|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ be normed spaces over a field $F$ and let $T: \mathscr{D}(T) \to Y$ be a linear operator with $\mathscr{D}(T) \subset X$. Then $T$ is a **bounded linear operator** if
>
> $$
> \exists c \in F \forall x \in \mathscr{D}(T): \|Tx\|_Y \leq c \|x\|_X.
> $$
In this case we may also define the set of all bounded linear operators.
> *Definition 5*: let $\mathscr{B}(X,Y)$ denote the set of bounded linear operators mapping from a vector space $X$ to a vector space $Y$.
We have the following theorem.
> *Theorem 3*: let $X$ and $Y$ be vectors spaces, the set of bounded linear operators $\mathscr{B}(X,Y)$ is a subspace of $\mathscr{L}(X,Y)$.
??? note "*Proof*:"
Will be added later.
Likewise, we may call $\mathscr{B}(X,Y)$ the space of bounded linear operators.
The smallest possible $c$ such that the statement in definition 4 still holds is denoted as the norm of $T$ in the following definition.
> *Definition 5*: the norm of a bounded linear operator $T \in \mathscr{B}(X,Y)$ is defined by
> and the norm of a bounded linear operator is a norm.
??? note "*Proof*:"
Will be added later.
Note that the second statement in lemma 2 is non trivial, as the norm of a bounded linear operator is only introduced by a definition.
> *Proposition 4*: if $(X, \|\cdot\|)$ is a finite-dimensional normed space, then every linear operator on $X$ is bounded.
??? note "*Proof*:"
Will be added later.
By linearity of the linear operators we have the following.
> *Theorem 4*: let $X$ and $Y$ be normed spaces and let $T: \mathscr{D}(T) \to Y$ be a linear operator with $\mathscr{D}(T) \subset X$. Then the following statements are equivalent
>
> 1. $T$ is bounded,
> 2. $T$ is continuous in $\mathscr{D}(T)$,
> 3. $T$ is continuous in a point in $\mathscr{D}(T)$.
??? note "*Proof*:"
Will be added later.
> *Corollary 1*: let $T \in \mathscr{B}$ and let $(x_n)_{n \in \mathbb{N}}$ be a sequence in $\mathscr{D}(T)$, then we have that
>
> 1. $x_n \to x \in \mathscr{D}(T) \implies Tx_n \to Tx$ as $n \to \infty$,
> 2. $\mathscr{N}(T)$ is closed.
??? note "*Proof*:"
Will be added later.
Furthermore, bounded linear operators have the property that
$$
\|T_1 T_2\| \leq \|T_1\| \|T_2\|,
$$
for $T_1, T_2 \in \mathscr{B}$.
??? note "*Proof*:"
Will be added later.
> *Theorem 5*: if $X$ is a normed space and $Y$ is a Banach space, then $\mathscr{B}(X,Y)$ is a Banach space.
??? note "*Proof*:"
Will be added later.
> *Definition 6*: let $T_1, T_2 \in \mathscr{L}$ be linear operators, $T_1$ and $T_2$ are **equal** if and only if
>
> 1. $\mathscr{D}(T_1) = \mathscr{D}(T_2)$,
> 2. $\forall x \in \mathscr{D}(T_1) : T_1x = T_2x$.
## Restriction and extension
> *Definition 7*: the **restriction** of a linear operator $T \in \mathscr{L}$ to a subspace $A \subset \mathscr{D}(T)$, denoted by $T|_A: A \to \mathscr{R}(T)$ is defined by
>
> $$
> T|_A x = Tx,
> $$
>
> for all $x \in A$.
Furthermore.
> *Definition 8*: the **extension** of a linear operator $T \in \mathscr{L}$ to a vector space $M$ is an operator denoted by $\tilde T: M \to \mathscr{R}(T)$ such that
>
> $$
> \tilde T|_{\mathscr{D}(T)} = T.
> $$
Which implies that $\tilde T x = Tx\; \forall x \in \mathscr{D}(T)$. Hence, $T$ is the resriction of $\tilde T$.
> *Theorem 6*: let $X$ be a normed space and let $Y$ be Banach space. Let $T \in \mathscr{B}(M,Y)$ with $A \subset X$, then there exists an extension $\tilde T: \overline M \to Y$, with $\tilde T$ a bounded linear operator and $\| \tilde T \| = \|T\|$.