Definition:Support Functional

Definition

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

Let $\struct {X^\ast, \norm \cdot_{X^\ast} }$ be the normed dual space of $X$.

Let $x \in X$.

We say that $f \in X^\ast$ is a support functional at $x$ if and only if:

$(1): \quad$ $\norm f_{X^\ast} = 1$

$(2): \quad$ $\map f x = \norm x_X$.

Also see

• Existence of Support Functional

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $20.1$: Existence of a Support Functional