mathematics-physics-wiki/docs/en/mathematics/functional-analysis/normed-spaces/linear-functionals.md

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.

??? note "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'.

??? note "Proof:"

Will be added later.