Norm on Bounded Linear Functional is Finite

Theorem

Let $V$ be a normed vector space.

Let $L$ be a bounded linear functional on $V$.

Let $\norm L$ denote the norm on $L$ defined as:

$\norm L = \inf \set {c > 0: \forall v \in V: \size {L v} \le c \norm v_V}$

Then:

$\norm L < \infty$

Proof

By definition of a bounded linear functional:

$\exists c \in \R_{> 0}: \forall v \in V: \size{L v} \le c \norm v_V$

Hence:

$\set {\lambda > 0: \forall v \in V: \size {L v} \le \lambda \norm v_V} \ne \O$

By definition: $\set {\lambda > 0: \forall v \in V: \size {L v} \le \lambda \norm v_V}$ is bounded below by $0$.

From the Greatest Lower Bound Property:

$\norm L = \inf \set {\lambda > 0: \forall v \in V: \size {L v} \le \lambda \norm v_V}$ exists.

We have:

\(\ds \norm L\)

\(\le\)

\(\ds c\)

Definition of Infimum

\(\ds \)

\(<\)

\(\ds \infty\)

because $c \in \R_{>0}$

The result follows.

$\blacksquare$