A new $2$-norm generated by bounded linear functionals on a normed space

Abstract

In a normed space $X$ with dimension dim$(X)\geq 2$, we introduce a new $2$-norm generated by bounded linear functionals on $X$ and investigate
its relationship with the $2$-norm defined by G\"{a}hler [Lineare $2$-normierte R\"{a}ume, Math. Nachr.]. We also derive a norm on $X$ from this $2$-norm and explore its relation with the norm on $X$.