Some remarks on continuous sublinear functionals on Banach spaces (Q2731026)

Let \((X,\|\;\|)\) be a real Banach space and NEWLINE\[NEWLINE(x,y)_i=\lim_{\hskip 1mm t\to 0_-} {\|y+tx\|^2 - \|y\|^2\over 2t}\cdot .NEWLINE\]NEWLINE First, by using Bishop-Phelps theorem, the author proves that: NEWLINENEWLINENEWLINEfor every continuous sublinear functional \(p: X\to {\mathbf R}\) and every \(\varepsilon>0\), there exists an element \(u_{p,\varepsilon}\in X\) such that \(p(x)\geq (x,u_{p,\varepsilon})_i-\varepsilon \|x\|\) for every \(x\in X\), NEWLINENEWLINENEWLINEand recovers some of his previous result on approximation of continuous linear functionals. Secondly, as an application of James theorem, he gets that: NEWLINENEWLINENEWLINE\(X\) is reflexive if and only if for every continuous sublinear functional \(p: X\to {\mathbf R}\) there is a \(u_p\in X\) such that \(p(x)\geq (x,u_p)_i\) for all \(x\in X\). NEWLINENEWLINENEWLINEIn particular, he recovers one of his results: NEWLINENEWLINENEWLINE\(X\) is smooth and reflexive if and only if for every \(f\in X^\ast\), there is a \(u_f\in X\) such that \(f(x)=[x,u_f]\) for all \(x\in X\), where \([x,y]=\lim\limits_{t\to 0} {\|y+tx\|^2 - \|y\|^2\over 2t}\cdot\).