Definition:Space of Bounded Linear Functionals

Definition

Let $\mathbb F$ be a subfield of $\C$.

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

We define the space of bounded linear functionals on $X$, written $\map B {X, \mathbb F}$, as the set of all bounded linear functionals on $X$.

Vector Space

Let $+$ denote pointwise addition of complex-valued functions.

Let $\circ$ denote pointwise scalar multiplication on linear functionals.

We say that $\struct {\map B {X, \mathbb F}, +, \circ}_{\mathbb F}$ is the vector space of bounded linear functionals on $X$.

Also see

• Definition:Space of Bounded Linear Transformations, of which this is a special case
• Definition:Normed Dual Space