Characterizations of inner product spaces: dimension 2 versus dimension \(\geq 3\) (Q2718474)

The author shows the conditions under which a real normed linear space \((X,\|\cdot\|)\) becomes a real inner product space (i.e. RIP Sp.). In order to describe the conditions dependent on the dimension of \(X\), the following two definitions are given:NEWLINENEWLINENEWLINE(i) BBJ-orthogonality \(x\perp_{BJ}z\) for \(x,z\in X\) is defined by \(\|x+\alpha z\|\geq\|x\|\) for \(\forall\alpha\in R\).NEWLINENEWLINENEWLINE(ii) Orthogonally additive mapping is the solution \(f: X\to Y\) of the Cauchy functional equation (i.e. CF Eq.) \(f(x+ z)= f(x)+ f(z)\) for \(\forall x,z\in X\) satisfying \(x\perp_{BJ}z\), where \((Y,+)\) is an Abelian group.NEWLINENEWLINENEWLINECondition K. Suppose that \(\dim X\geq 3\). If there exists a projection \(P\) satisfying \(P(X)= M\) and \(\|P\|\leq 1\) for any closed subspace \(M\), \(X\) is a RIP Space.NEWLINENEWLINENEWLINECondition DJ. Suppose that \(\dim X\geq 3\). If and only if \(\perp_{BJ}\) is symmetric, \(X\) is a RIP Space.NEWLINENEWLINENEWLINECondition RS. Suppose that \(\dim X\geq 2\). If not all the real-valued solutions of the CF Eq. are additive, \(X\) is a RIP space.NEWLINENEWLINENEWLINEThe author contributes to give the Condition RS.