Year: 2026
Type: preprint
Abstract: Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\le p\le\infty$, we equip $C^1(I)$ with the norm \[ \Vert{f}\Vert_{(p)} := \begin{cases} \left(|f(0)|^p+\Vert{f'}\Vert_{I}^p\right)^{1/p}, & 1\le p<\infty, \\ \max\{|f(0)|,\Vert{f'}\Vert_{I}\}, & p=\infty, \end{cases} \qquad (f\in C^1(I)), \] where $\Vert{\cdot}\Vert_I$ denotes the supremum norm on $I$. Let $\mathcal{S}_p^+:=\{f\in C^1(I):f(0)\ge0,\,f'\ge0,\,\Vert{f}\Vert_{(p)}=1\}$ be the positive unit sphere of $C^1(I)$. A map $T:\mathcal{S}_p^+\to \mathcal{S}_p^+$ is called norm-additive if \[ \|T(f)+T(g)\|_{(p)}=\|f+g\|_{(p)} \qquad (f,g\in \mathcal{S}_p^+). \] For $11$ is essential.
Source: arXiv (Cornell University)
Authors: Ezumi, Kazuki, Miura, Takeshi
Institution: Niigata University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220436759