7 problems

Let $S$ be an incomplete inner product space, and let $F(S)$ denote the orthogonally closed subspaces of $S$. A finitely additive state on $F(S)$ is a state that is additive on ort…

Let $X$ be a real normed space with unit sphere $S$. For $0<\rho<1$, say that $X$ fulfils property (P-$\rho S$) when … Here, an inner product space is a real normed space whose nor…

Let $(X,\|\cdot\|)$ be a complex normed space, and let $\langle\cdot\mid\cdot\rangle$ denote its product from the defining construction. Phase-linearity characterization. For all…

Let $X$ be a finite-dimensional real smooth normed linear space with $\dim X\geq 3$. A finite set $\{x_1,\ldots,x_n\}$ is strongly orthonormal in the sense of Birkhoff–James if, fo…

Phase-covariance characterization. This identity holds for all $\vec x,\vec y\in X$ if and only if the product $\langle\cdot\mid\cdot\rangle$ is an inner product, so that…

Let $(X,\|\cdot\|)$ be a complex normed vector space. Define its deformation by … where $\langle\vec a\mid\vec b\rangle$ is the product induced by the norm through the polarization…

For positive real numbers $\alpha,\beta,\gamma,\delta$ satisfying … let $\mathsf{NORM}_{\rho}$ denote the class of normed spaces having angle $\angle_{\rho}$, and let…