Linear functionals

Definition 1: a linear functional \(f\) is a linear operator with its domain in a vector space \(X\) and its range in a scalar field \(F\) defined in \(X\).

The norm can be a linear functional \(\|\cdot\|: X \to F\) under the condition that the norm is linear. Otherwise, it would solely be a functional.

Definition 2: a bounded linear functional \(f\) is a bounded linear operator with its domain in a vector space \(X\) and its range in a scalar field \(F\) defined in \(X\).

Dual space

Definition 3: the set of linear functionals on a vector space \(X\) is defined as the algebraic dual space \(X^*\) of \(X\).

From this definition we have the following.

Theorem 1: the algebraic dual space \(X^*\) of a vector space \(X\) is a vector space.

Proof:

Will be added later.

Furthermore, a secondary type of dual space may be defined as follows.

Definition 4: the set of bounded linear functionals on a normed space \(X\) is defined as dual space \(X'\).

In this case, a rather interesting property of a dual space emerges.

Theorem 2: the dual space \(X'\) of a normed space \((X,\|\cdot\|_X)\) is a Banach space with its norm \(\|\cdot\|_{X'}\) given by

\[ \|f\|_{X'} = \sup_{x \in X\backslash \{0\}} \frac{|f(x)|}{\|x\|_X} = \sup_{\substack{x \in X \\ \|x\|_X = 1}} |f(x)|, \]

for all \(f \in X'\).

Proof:

Will be added later.