1.1. Operators on Banach spaces

In this section, let \(X\) and \(Y\) be Banach spaces.

Definition 1.7 (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 \(\langle F, x \rangle_{X, Y}\).

Definition 1.8 (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.

Theorem 1.1

If \(Y\) is a Banach space, then \(\Lcal(X,Y)\) is also a Banach space.

Definition 1.9 (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 \(X\).

Theorem 1.2 (Riesz representation theorem)

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

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

1.1.1. Derivatives of operators on Banach spaces

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

Definition 1.10

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 \(F\).