# Operators on Banach spaces

In this section, let $X$ and $Y$ be Banach spaces.

````{prf:definition} Linear operator
:label: def-linear-operator

A mapping $F: X \to Y$ is a *linear operator* if

$$
F \left[ \lambda x + \mu y \right] = \lambda F x + \mu F y \qquad \forall \, x, y \in X, \ \lambda, \mu \in \Rbb.
$$
````

We write the application of the operator $F$ on $x \in X$ as $F(x)$ or, equivalently, as $\langle F, x \rangle_{X, Y}$.

````{prf:definition} Bounded operator
:label: def-bounded-operator

A linear operator $F: X \to Y$ is *bounded* if

$$
\|F\|_{X, Y} := \sup \left\{ \| F x \|_Y \ : \ \|x\|_X \le 1 \right\} < \infty.
$$

The space of all linear bounded operators from $X$ to $Y$ is called $\Lcal(X,Y)$. Note that $(\Lcal(X,Y), \| \cdot \|_{X, Y})$ is a normed vector space.
````

````{prf:theorem}
If $Y$ is a Banach space, then $\Lcal(X,Y)$ is also a Banach space.
````

````{prf:definition} Dual space
:label: def-dual-space

1. A bounded linear operator $x^*: X \to \Rbb$ is called a *bounded linear functional on $X$*.
2. We write $X^*$ to denote the collection of all bounded linear functionals on $X$. $X^*$ is also called the *dual space of $X$*.

For any $x \in X$, we use the notation

$$
\langle x^*, x \rangle_{X^*, X} = x^* (x) \in \Rbb.
$$

The symbol $\langle \cdot , \cdot \rangle_{X^*, X}$ is also denoted *pairing of $X^*$ and $X$*.
````

````{prf:theorem} Riesz representation theorem
:label: thm-riesz-representation

Let $H$ be a real Hilbert space with inner product $(\cdot,\cdot)$. Then, $H^*$ can be canonically identified with $H$. That is, for each $x^* \in H^*$, there exists a unique $x \in H$ such that

$$
\langle x^*, y \rangle_{X^*, X} = (x, y) \qquad \forall y \in H.
$$
````

## Derivatives of operators on Banach spaces

Differentiability of operators on Banach spaces are a little more complicated and involved than normal functions.

````{prf:definition}
:label: def-differentiability-banach

Let $F: U \subset X \to Y$ with $X, Y$ Banach spaces and $U \neq \emptyset$.

1. The *directional derivative* $dF$ evaluated at $x \in U$ in the direction $h \in X$ is defined as

    $$
    dF (x; h) = \lim_{\epsilon \to 0} \frac{F(x + \epsilon h) - F(x)}{\epsilon} \in Y
    $$

    if the limit exists. In such case, $F$ is called *directionally differentiable* at $x$.

2. $F$ is *Gateaux differentiable* at $x \in U$ if $F$ is directionally differentiable at $x$ and the mapping $F'(x): X \to Y$ defined as

    $$
    \langle F'(x), h \rangle_{X, Y} = dF(x; h)
    $$

    is bounded and linear, i.e. $F'(x) \in \Lcal(X, Y)$. The operator $F'$ is called *(Gateaux) derivative of $F$*.
````