Definition:Space of Bounded Linear Functionals/Vector Space

Definition

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

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

Let $\map B {X, \mathbb F}$ be the space of bounded linear functionals.

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

• Space of Bounded Linear Functionals with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space shows that $\struct {\map B {X, \mathbb F}, +, \circ}_{\mathbb F}$ is indeed a vector space.