Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain/Corollary 1

Corollary to Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain

Let $\Bbb F \in \set {\R, \C}$.

Let $\struct {X, \norm \cdot_X}$ be a normed vector space over $\Bbb F$.

Let $\map D {f_0}$ be a linear subspace of $X$.

Let $f_0 : \map D {f_0} \to \Bbb F$ be a bounded linear functional.

Then there exists a unique bounded linear functional $f : \map D f \to \Bbb F$ extending $f_0$ to $\map D f = \paren {\map D {f_0} }^-$.

Proof

This is a direct consequence of Bounded Linear Transformation to Banach Space has Unique Extension to Closure of Domain and:

Real Number Line is Banach Space if $\Bbb F = \R$

Complex Plane is Banach Space if $\Bbb F = \C$.

$\blacksquare$