Characterization of von Neumann-Boundedness in Normed Vector Space

Theorem

Let $\GF \in \set {\R, \C}$.

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

Let $U \subseteq X$.

Then $U$ is von Neumann-bounded if and only if there exists $M > 0$ such that:

$\norm x < M$

for all $x \in U$.

Proof

From Normed Vector Space is Hausdorff Locally Convex Space, $\struct {X, \norm {\, \cdot \,} }$ can be viewed as the Hausdorff locally convex space $\struct {X, \norm {\, \cdot \,} }$.

The result is then immediate from Characterization of von Neumann-Boundedness in Hausdorff Locally Convex Space.

$\blacksquare$