ON THE QUASI-INNER PRODUCT SPACES

Abstract

A quasi-inner product space \(X\) (q. i. p. space) is strictly convex. If a sequence \((x_n)\) converges weakly to \(x_0\) (\(x_n,x_0\in X\), \(x_n\to x_0\)) and \(\|x_n\|\to\|x_0\|\), then \(\|x_n-x_0\|\to 0\). The orthogonality relation \(\perp^g\) defined by (7), is uniquely resolvable, i.e. there exists a unique \(a\in\mathbb{R}\) such that \(x\perp^g(ax+y)\) (\(\|x\|\cdot\|y\|\ne 0\)). Under certain conditions the vector \(-ax\) is the best approximation of the vector \(y\) with the vectors from \([x]:=\operatorname{span}\{x\}\). In regard to the relation \(\perp^g\) in a q.i.p. space, the lengths of the diagonals of the parallelogram are equal and the diagonals are perpendicular iff this parallelogram is a rectangle. A q.i.p. space i an inner product space iff (27) holds.